|  MIML: Multiplex Image Machine Learning for High Precision Cell
Classification via Mechanical Traits within Microfluidic Systems
---|---
|  Khayrul Islam,a Ratul Paul,a Shen Wang, a and Yaling Liu ∗a,b
|  Label-free cell classification is advantageous for supplying pristine cells
for further use or examination, yet existing techniques frequently fall short
in terms of specificity and speed. In this study, we address these limitations
through the development of a novel machine learning framework, Multiplex Image
Machine Learning (MIML). This architecture uniquely combines label-free cell
images with biomechanical property data, harnessing the vast, often
underutilized morphological information intrinsic to each cell. By integrating
both types of data, our model offers a more holistic understanding of the
cellular properties, utilizing morphological information typically discarded
in traditional machine learning models. This approach has led to a remarkable
98.3% accuracy in cell classification, a substantial improvement over models
that only consider a single data type. MIML has been proven effective in
classifying white blood cells and tumor cells, with potential for broader
application due to its inherent flexibility and transfer learning capability.
It’s particularly effective for cells with similar morphology but distinct
biomechanical properties. This innovative approach has significant
implications across various fields, from advancing disease diagnostics to
understanding cellular behavior.

##

††footnotetext: a Lehigh University, Mechanical Engineering, Bethlehem, 18015,
USA; E-mail<EMAIL_ADDRESS>b Lehigh University, Bio
Engineering, Bethlehem, 18015, USA

##  1 Introduction

Identifying and sorting target cells from heterogeneous populations
constitutes a crucial initial step in numerous biological, biotechnological,
and medical applications 1, 2. Following sorting, these cells may undergo
detailed analysis, probing their proteomic, transcriptomic, or genetic
identities and functions 3, 4, 5. Alternatively, they can be utilized for
regenerative medicine applications, such as transplantation into patients 6,
7. Cell sorting is traditionally performed based on molecular labels 8, 9, 10.
However, sorting methods leveraging intrinsic properties, such as cell size or
deformability, have also been demonstrated 11, 12, 13.

Fig. 1: MIML Inferencing Process and Cell Analysis, (a) Schematic of the cell
data collection and subsequent classification, (i) Biomechanical data
collection, (ii) Preparation of cell samples, (iii) Cell transition through a
narrow channel, (iv) MIML inferencing using cell imagery and textual data, (b)
Experimental cell imagery, (i) Cell length prior to compression, (ii) Cell
length while being compressed, (iii) Snapshot of a cell positioned centrally
within the narrow channel, (c) Temporal velocity profile of cells, (d)
Normalized cell progression through the squeezing channel.

Nonetheless, it is imperative to acknowledge the limitations that emerge from
the current approaches. While the high accuracy resulting from fluorescent
methodologies is obtainable, it is not without its drawbacks14, 15, 16, 17,
18. The process of fluorescence labeling is both time-consuming and costly.
Furthermore, fluorescent markers can interfere with cellular function and
physiology, potentially altering their natural states and behaviors, thereby
compromising the integrity of the research results[see supplementary note 2
for drawbacks of fluorescent-based sorting]. Particularly when applying label-
free bright field imaging techniques for cell detection, a decrease in
classification accuracy is observed, especially concerning visually
indistinguishable cells. Recently, machine learning has begun to significantly
reshape the contours of biomedical imaging, introducing a revolutionary level
of precision and analytical depth19, 20, 21, 22, 23, 24, 25. The majority of
machine learning-based on cell classifications have traditionally hinged on
either image-based methods or the extraction of specific cell features 26, 27,
28, 29, 30, 31, 32. Only a handful of studies have explored the intersection
of label-free cell classification and machine learning 19, 33, 34. This
emerging field is crucial for developing refined cell classification methods
for visually indistinguishable cells. Addressing this need has the potential
to transform our understanding of cellular dynamics. An advanced, label-free
classification could significantly advance cell biology and enhance biomedical
imaging, paving the way for less invasive studies and new breakthroughs in
cell classification.

In our endeavor to refine cell classification methodologies, we developed a
novel machine learning architecture called Multiplex Image Machine Learning
(MIML). This architecture is specially crafted to categorize visually similar
cells, leveraging both label-free bright field images and intrinsic cellular
mechanical properties as primary input features. The significance of including
mechanical properties lies in their inherent ability to reflect cellular
behaviors and states. MIML is implemented in a use case involving the
classification of Tumor cells (HCT116s as an example) and White Blood Cells
(WBCs) achieving a significant accuracy of 98.3% with a substantial
improvement of $\sim 8\%$ over pure image-based classification. With these
insights and results in place, the core achievements of our study are
summarized in the following research highlights:

  1. 1.

Architecture: An entirely new machine learning architecture is designed and
introduced. This framework uniquely combines both image and cell mechanical
properties to predict tumor cell types with high accuracy. Such an approach
circumvents the necessity for fluorescent labeling, thus eliminating the
associated drawbacks.

  2. 2.

Interpretability: Through comprehensive feature analysis and activation layer
visualization, we have achieved an interpretable AI model for tumor cell
analysis. This clarity facilitates a deeper understanding and yields higher
accuracy in cell classification.

  3. 3.

Transfer Learning: Our research successfully demonstrated a novel approach
that utilizes a combination of cell images and distinct mechanical properties
for the classification of cells that are visually indistinguishable. The
potential applicability of this methodology is extensive, and it could be
generalized to classify an array of cell types that, despite exhibiting visual
similarities, bear distinguishable cellular properties.

##  2 Results and Discussion

The schematic representation of the standard procedure for cell detection via
MIML is embodied in Figure 1(a). The subsequent sections are dedicated to a
comprehensive examination of various classification models apt for numerical
data. We also delve into traditional Convolutional Neural Network (CNN) models
designed for image classification. Finally, we introduce our custom MIML
model. The MIML model implements an innovative approach to predict cell type.
It efficiently unifies images captured in real time as cells traverse through
a narrow channel. The mechanical properties of these cells are evaluated
through image processing techniques. This combination of visual and mechanical
data allows for a more accurate and nuanced understanding of the cell type,
thereby elevating the performance of cell detection and classification to
unprecedented levels.

Fig. 2: Examination of cellular biomechanical properties, (a-c) feature
scatter plots that demonstrate the variability of inherent cellular
attributes, (d) illustration of correlations between various features, as
portrayed in a heatmap, accompanied by associated p-values (f) cross-
validation utilized in this study, illustrating a scenario with four stacks
used for training and one stack reserved for testing.

The effectiveness of the MIML model is demonstrated through an example
application of differentiating tumor cells from white blood cells. Detection
of circulating tumor cells in blood samples is important for early cancer
diagnosis and monitoring of tumor progression 35, 36, 37 [cite a few papers,
including ours]. Label-free detection of circulating tumor cells without
fluorescence labeling and antibodies is gaining popularity because they can be
applied to a wide range of cancer types without a tedious labeling process 31,
38, 39 [cite a few papers, including ours]. In this paper, we used a model
circulating tumor cell sample by mixing HCT116 cells with white blood cells.
HCT116 cells are a commonly used human colon cancer cell line, derived from a
patient with colorectal carcinoma. Their genomic stability and well-
characterized nature make them pivotal in oncological research. The goal is to
demonstrate that a combination of image features and mechanical features in an
integrated machine learning framework leads to higher classification accuracy
than any of these features alone.

###  2.1 Composition of training and validation sets

In our study, we utilize two distinct forms of data for the purpose of
training our models. The first, textual data, is provided in a CSV format and
serves as the training dataset for our classification models. The second,
image data, is deployed in the training of our CNN model. In the interest of
robust evaluation, the data is partitioned into three distinct sets, namely,
training, testing, and validation. The role of the testing data is to
continuously assess the model performance throughout the course of training,
whereas the validation data is reserved for an evaluation subsequent to the
completion of the training process. This approach is pivotal in directing our
model to recognize critical parameters, thereby aiding in the attainment of
enhanced validation performance. Additionally, this method serves as a
preventive measure against potential overfitting that comes from model
learning parameters that result in high performance on the training and
testing data but fail to generalize in overall application 40, 41

To get the mechanical properties of the cells, we have designed and fabricated
a microfluidic device with a narrow channel smaller than the cell size. While
the cells pass through such narrow channels, they will experience large
deformation, leading to different translocation speeds and times. For each
individual cell navigating through the narrow channel, we have captured two
images - one at the beginning and another at the termination of the squeezing
process 1(b)(iii). These images serve to train our CNN model, while an
additional training data point per cell is employed for training our
classification models. The total training data comprises 2521 entries, of
which 1156 correspond to White Blood Cells (WBC) and 1365 to HCT-116 cells.
For the purpose of training the CNN model, we possess a total of 5042 images,
segregated into 2312 images of WBC and 2730 of HCT-116. The data corpus has
been partitioned into training and testing sets with a 4:1 ratio. The larger
portion (80%) is allocated for training and validating purposes, while the
remainder (20%) is set aside for testing. This training set is further
subdivided into five subsets for the implementation of cross-validation.
During each iteration of the cross-validation, one subset is reserved for
validating while the remaining four subsets are employed for training Figure
2(f). This meticulous data partitioning and utilization strategy underpins our
methodical approach toward robust model training and performance validation.

###  2.2 Detection of cell classes by classification model

Image-derived metrics such as deformation, defined as the divergence from a
perfectly circular shape, offer insight into the mechanical properties of the
objects being measured 42, 43. This deformation, coupled with maximum velocity
and transition time, emerges as a crucial parameter set for effective cell
classification. We analyzed three distinct features of a cell navigating a
narrow channel, namely, the cell’s deformation Index $(DI)$, the Transition
Time $(TT)$ taken by the cell to traverse the narrow channel, and the maximum
velocity $v_{\text{max}}$ achieved by the cell within the channel’s narrow
region1(b)(iii). The DI of the cell is quantified using the following formula:

$DI=\frac{(a-b)}{(a+b)}$ (1)

In this equation, $a$ and $b$ represent the major and minor axes of the
considered cell, respectively. The deformation index adopts a scale from 0 to
1, with 0 representing a flawless circle, thus indicating no deformation, and
1 corresponding to the utmost conceivable deformation, given an assumed minor
axis of zero44. As the deformation index values come pre-normalized, it is
crucial to extend this normalization to the remaining pair of features -
transition time and maximum velocity. By doing so, we amplify our model’s
generalization potential and encourage its adaptability across different
contexts through transfer learning. Therefore, we ensure these features also
conform to a 0-1 scale and proceed to scrutinize their correlation.

Fig. 3: Analysis of Classification Models. (a-e)(i-v) showcases the predictive
capabilities of each classification model for varying feature combinations.

Figure 1 provides a detailed representation of the data acquisition
methodology, complemented by the microfluidic channel employed for cellular
deformation. The velocity, as delineated in 1(c), is normalized in accordance
with the flow velocity. Meanwhile, Figure1(d) depicts the position of the
cell’s centroid, which is presented in values normalized relative to the
cell’s dimensions. This figure effectively captures the trajectory of the cell
within the microfluidic channel. Figure 2(a-c) illustrated scatter plots
capturing the relationship between each pair of feature sets under study. A
notable observation from Figure 2(a) is that HCT-116 cells consistently
exhibit shorter transition times and greater degrees of deformation as
compared to White Blood Cells (WBCs). The underlying reason for this
distinction lies in the inherent biophysical properties of these cell types.
HCT-116 cells are characterized by a lower stiffness 45, 46, which confers
upon them a higher degree of flexibility. This malleability allows these cells
to adapt their shape more readily in response to external pressures,
consequently enabling a swifter transit through narrow spaces. This quality
not only enhances their overall velocity but also results in a decreased
transition time, albeit at the cost of experiencing greater deformation. The
same reasoning can explain the lower maximum velocity of the WBCs due to its
reduced deformability. The relative stiffness of WBCs impairs their ability to
modify their shape optimally to navigate the physical constraints of the
narrow channel. This limitation subsequently slows down their transit time, as
they lack the shape flexibility needed to maintain higher velocities.

Fig. 4: Analysis of Neural Network Model, (a)the architectural design of the
neural network dedicated to predicting cell classes, (b) the confusion matrix,
outlining the model performance against the testing dataset from the neural
network, (c) computed accuracy across all models.

Advancing from these observations, Figure 2(d) provides a graphical depiction
of the correlation matrix in the form of a heatmap, complemented by
accompanying p-values. Each p-value is a statistical metric that measures the
strength of evidence for rejecting the null hypothesis in the context of
evaluated features. Notably, all p-values displayed are less than 0.05,
indicating strong evidence against the null hypothesis. This suggests the
correlations observed are statistically significant and unlikely due to random
chance. Even more compelling are two specific correlations (R1 and R3), with
p-values $\sim 0$, implying an exceedingly strong level of statistical
significance. To further investigate the training features, we plotted
regression coefficients along with measured error bars. These coefficients
provide a quantitative measure of the rate of change in one variable
(dependent) due to a one-unit change in another variable (independent).
Remarkably, the third relationship (R3), which is between maximum velocity and
transient time, showcases a high regression coefficient of $\sim 0.96$. This
high value suggests a robust association between these two variables, a
finding that aligns intuitively with our understanding. As expected, the time
needed for a cell to traverse through a narrow channel is inversely
proportional to its velocity, an increase in velocity leads to a decrease in
transit time, hence the strong relationship.

Figure 2(f) presents our implementation of cross-validation, a critical
technique in statistical modeling, where we designated four stacks for
training and one stack for validation purposes. Cross-validation is
fundamental to the robustness of machine learning models as it mitigates
overfitting, enabling us to assess how well our model generalizes to unseen
data. By partitioning our data into separate training and validation stacks,
we can train the model on one set of data and then validate it on a completely
separate set. This not only provides an unbiased evaluation of the model’s
performance but also ensures that our findings are not mere artifacts of our
training data, thus promoting the generalizability and reliability of our
model. Furthermore, following the exhaustive cross-validation procedure, we
scrutinized the model’s performance utilizing a separate testing dataset. This
dataset was not part of either the training or validation phase of the cross-
validation process. This evaluation strategy allowed us to further test the
model’s performance on unfamiliar data, thereby helping us verify that the
model does not produce biased results based on its training and validation
data40, 47.

Our study delved into the exploration of the predictive efficacy of several
classification models, specifically Logistic Regression (LR), Support Vector
Machine (SVM), Decision Tree (DT), Random Forest (RF), and K-Nearest Neighbors
(KNN). Each of these models provides insights, as they are grounded in
distinct computational approaches and theoretical underpinnings. LR, for
instance, relies on statistical analysis to estimate probabilities, while the
SVM employs geometric principles to maximize the margin between classes. DT
and RF are built upon hierarchical structures that aim to split data into
distinct subsets based on feature characteristics. Lastly, the KNN model
classifies new instances based on their proximity to existing instances in the
feature space48, 49. To get a nuanced understanding of the predictive power of
these models, we employed each pair of features in our dataset to train these
models. The results of these model predictions, excluding those from the
Neural Network (NN), are visually represented in Figure 3(a-e)(i-v). This
comparative analysis not only helps us understand how different models
interpret and predict the relationships between specific feature pairs but
also serves as an invaluable resource for refining hyperparameters for model
optimization. Through the systematic scrutiny of diverse models and feature
combinations, we have obtained a comprehensive perspective of the predictive
landscape inherent in our data. We further extended our exploration to the
prediction capabilities of NNs. Figure 4(a) delineates the architecture of the
NN model that was employed in predicting cell types based on the features
previously discussed. This architecture obtained through our ablation study
(see supplementary material note 3 for detail) constitutes a simple yet
effective network comprising two hidden layers containing 32 and 16 neurons,
respectively. The first three input neurons are engaged in processing the
input, while the final two neurons are tasked with generating the predicted
class. To further quantify the performance of our model, we have also included
a confusion matrix in Figure 4(b), which was derived from the testing dataset.
A confusion matrix is a critical tool in machine learning that allows for a
detailed assessment of a classifier’s performance. It provides insights into
not only the model’s accuracy but also the nature of errors it makes,
delineating false positives and negatives as well as true positives and
negatives. This granular evaluation assists in identifying areas for model
improvement and enhances our understanding of its predictive capabilities.

Fig. 5: Analysis of Convolution Neural Network(CNN). (a)Sample images from
training and validation dataset (b) Training and testing accuracy while
training (c) Feature map visualization for WBC (d) Grad-CAM visualization for
WBC and HCT-116 (e) the confusion matrix, outlining the model performance
against the training and validation dataset from the CNN. (f) illustration of
our CNN architecture along with the TSNE plot generated from latent space.

In line with this performance evaluation, Figure 4(c) showcases a comparative
analysis of the accuracy of each evaluated model. Accuracy, in this context,
refers to the proportion of correct predictions made by the model relative to
the total number of test datasets. This measure serves as a fundamental metric
in evaluating the overall performance of classification models. From the
visual representation, it is evident that the Neural Network (NN) model
exhibits best performance compared to the other models we tested. Its
demonstrated ability to more accurately classify cell types based on the
feature sets used instills confidence in its predictive power. Given its
performance, we decided to adopt the NN model for further exploration and
classification of cells in our study. This decision is grounded not only in
the NN model’s accuracy but also in its capacity for non-linear classification
and its inherent ability to learn complex patterns, making it an ideal choice
for our subsequent investigation into cell classification.

###  2.3 Detection of cell classes by Convolutional neural network

The CNN we used can be divided into two parts: a convolutional feature
extraction part (called ‘Encoder’) followed by the fully connected layers
classifying the input based on the features (Figure 5(g)). The encoder is
structured with an initial convolutional layer, succeeded by four progressive
stages, each consisting of multiple blocks. These blocks are critical to the
architecture, as they incorporate shortcut connections that perform identity
mapping, with their outputs added to the outputs of the stacked layers. The
shortcut connection is critical to solving the vanishing gradient problem, a
common obstacle encountered during the training of deep NNs50. The initial
layer is a 7x7 convolutional layer with a stride of 2, followed by batch
normalization and a ReLU activation function, and finally max pooling. The
four subsequent stages contain two blocks each, with the number of
convolutional filters doubling at every stage, beginning from 64 filters in
the first stage. Downsampling is performed by convolving with a stride of 2 in
the first layer of the 2nd, 3rd, and 4th stages, excluding the shortcut
connections, where 1x1 convolutions are applied to match the dimensions. Each
of these convolutional layers is succeeded by batch normalization and a ReLU
activation function. Following the final stage, there is a global average
pooling layer and a fully connected layer that leads to the final
classification output. The design of our CNN model, with its specialized
blocks and skip connections, provides an efficient way to train deep networks
by facilitating the propagation of gradients throughout the entire network.
The accuracies obtained from the training and validation datasets have been
visualized in Figure 5(b). As the figure illustrates, following 40 epochs of
training, we observe a steady fluctuation in accuracy levels. This consistency
in fluctuation suggests that the model reaches a relatively stable state of
learning after the 40th epoch, indicating diminishing returns from further
training. It’s critical to highlight validation accuracy is almost the same as
the training accuracy, implying that the model maintained a balance between
learning from the training set and generalizing to the validation set. This
observation is crucial in asserting the model’s capacity to avoid overfitting,
hence providing a reliable and robust solution for the task at hand.

Figure 5(c) offers a detailed graphical representation of the first
convolutional layer’s feature map for WBC, incorporating all 64 filters of
this layer. This feature map is an integral aspect of our study as it captures
the distinctive features that the convolutional layer has learned to identify.
Each of the 64 filters in this layer has learned to recognize different
characteristics of WBC. For instance, some might be specializing in detecting
the contours, some may focus on textural information, while others might be
zeroing in on more complex patterns. Visualizing these feature maps allows us
to gain an understanding of the underlying mechanics of our model—what exactly
it is picking up from the WBC images. By investigating these visualizations,
we are essentially interpreting the model’s learning process, which allows
refining our model and augmenting its overall performance in the process51.
Equally important is the role of Gradient-weighted Class Activation Mapping
(Grad-CAM) in model interpretability52. In Figure 5(d), we illustrate its
application, showcasing Grad-CAM for the penultimate and last convolutional
layers, with a particular focus on HCT-116 and WBC. The heat maps generated
through the Grad-CAM are superimposed over the original images to provide a
lucid understanding of the model’s focus during its learning process. From the
representation, it is evident that the penultimate convolutional layer is
casting a broad net, capturing a substantial amount of background information,
yet the primary emphasis remains on the cell structure. This layer acts as a
broad filter, capturing both the cell and its surrounding context, which can
be critical in many image recognition tasks. As the model progresses to the
last layer, it significantly refines its focus. It zeroes in predominantly on
the areas of the images that encapsulate the cell, displaying an acute
understanding of the cell’s morphology. This targeted approach underscores the
layer’s role in the identification of cell types based on their distinct
morphological features. For the assessment of our model, we utilized the
Receiver Operating Characteristic (ROC) curve, a crucial graphical tool for
the evaluation of binary classification models, as illustrated in Figure 5(e).
By graphically contrasting the true positive rate (TPR) against the false
positive rate (FPR) at various decision thresholds, the ROC curve serves as a
potent tool to measure the efficacy of our classification model53, 54. In this
instance, the ROC curves have been generated separately for both the training
and testing datasets. Notably, the Area Under the Curve (AUC), a crucial
performance metric for the classifier, manifests a remarkable consistency for
both our training and validation datasets (Training area $\sim$ testing area =
0.01). This consistency in the AUC values implies that our model exhibits no
signs of overfitting - a common complication in machine learning where models
tailor themselves too closely to the training data, compromising their
capacity to generalize on unseen data. Instead, our model illustrates a
balance between learning and generalizability, an unavoidable attribute in
practical applications.

Figure 5(f) offers a visualization of the predictions from our trained model
in the form of a confusion matrix drawn separately for both the training and
validation datasets. The results reveal a commendable level of accuracy for
both datasets. The model’s performance on the training data showcases an
accuracy of $\sim 91.81\%$, demonstrating its effective learning from the
given samples. Concurrently, the model has exhibited commendable performance
on the validation dataset, attaining an accuracy of $\sim 90.05\%$. The
validation accuracy is of particular importance as it indicates how well our
model is likely to perform on unseen, real-world data. The close proximity of
these accuracy values suggests a well-balanced model that has avoided
overfitting, demonstrating robust learning from the training data while still
maintaining the ability to generalize effectively to new data. We extracted
the features predicted by our model from the latent space and visualized them
via a t-Distributed Stochastic Neighbor Embedding (t-SNE) plot to unfold the
high-dimensional data narrative. t-SNE, a robust machine learning algorithm,
excels in the visualization of high-dimensional data55. It converts the
similarities among data points into joint probabilities, endeavoring to
minimize the Kullback-Leibler divergence between these joint probabilities in
the low-dimensional embedding and the original high-dimensional data. This
powerful technique provides a pathway to visualize the high-dimensional data
captured in our CNN’s latent space in a 2D format, offering an easily
interpretable perspective. In our study, the latent space of our CNN model
stored high-dimensional representations of the inputs, which encapsulated the
abstract features that the model had learned. Transposing these
representations into a t-SNE plot allowed us to take this complex, high-
dimensional information and present it in a comprehensible, visually coherent
format. Upon inspecting the t-SNE visualization, we observed a notable overlap
between the clusters representing White Blood Cells (WBC) and HCT-116. This
overlap suggests a visual similarity between these two cell types that the
image-based CNN model has difficulty differentiating. This insight emphasizes
the limitations of image-only models like CNNs in distinguishing intricate
cellular characteristics and underscores the potential need for integrating
other forms of data to improve cell differentiation performance.

###  2.4 Multiplex Image machine learning for cell detection

To enhance the accuracy of cell classification, we have developed a novel
architectural model, aptly termed the Multiplex Image Machine Learning (MIML)
Architecture. This cutting-edge model is capable of processing both image and
text data concurrently, enabling it to predict cell types based on a more
comprehensive set of input data. The MIML Architecture’s underlying mechanics
involve the fusion of image and text features containing cell mechanical
properties as described in the results section above. Specifically, after
processing the cell images through CNN, the resultant latent space is combined
with the output from the traditional NN, which processes the textual
information. This integration happens at a Fully Connected (FC) layer,
ensuring that the combined features from both modalities are effectively
utilized for the final prediction in cell classification. Our MIML model
seamlessly integrates a CNN with a traditional NN for advanced performance.
The strength of this integrated model lies in its ability to seamlessly handle
and interpret both cell images and associated textual information (mechanical
properties) regarding cell properties. As such, it encapsulates a broader
perspective of cellular data, facilitating a more nuanced understanding and
classification of the cells. Our empirical results underline the efficacy of
the MIML model, yielding significantly higher accuracy levels in comparison to
standalone implementations of the CNN or NN models. This improvement
underscores the potential of leveraging multi-modal data – incorporating both
image and text – to substantially enhance the performance of cell
classification tasks in machine learning applications.

Fig. 6: Model Performance Evaluation, (a)-(e) present the comparative
assessment of Accuracy, Precision, Recall, F1 score, and Area Under the
Receiver Operating Characteristic (AUROC) across various models, (f) showcases
the training, testing, and validation accuracy specific to the Multi-Instance
Multi-Label (MIML) model, (g) displays the confusion matrix pertaining to both
training and testing datasets within the MIML model.

In our exploration of the efficacy of various machine learning models,
including the MIML model, we investigate several key performance indicators.
Specifically, we focused on metrics such as accuracy, precision, recall, the
F1 score, and the area under the Receiver Operating Characteristics (AUROC) as
shown in Figure6(a-e). Our evaluation framework was constructed around the
aggregate count of true positives (TP), true negatives (TN), false positives
(FP), and false negatives (FN) that were recorded during the model’s
predictions. These components form the foundation for our performance metrics,
and their careful consideration is vital in the detailed dissection of our
model’s performance eq.(2 \- 5).

Fig. 7: Overview of MIML Cell Classification: (a) Schematic representation of
the MIML architecture for cell class prediction, (b) TSNE visualization
derived from the latent space, (c)(i-viii) Randomly chosen images from
selected patches, with (i-iv) representing WBC and (v-viii) depicting HCT116.

$\begin{split}\text{Accuracy}&=\frac{TP+TN}{TP+TN+FP+FN}\end{split}$ (2)

$\begin{split}\text{Precision}=\frac{TP}{TP+FP}\end{split}$ (3)

$\begin{split}\text{Recall}=\frac{TP}{TP+FN}\end{split}$ (4)

$\begin{split}F1=\left({\text{Precision}}^{-1}+{\text{Recall}}^{-1}\right)=\frac{2TP}{2TP+FP+FN}\end{split}$
(5)

Accuracy is a fundamental measure providing a clear overview by accounting for
the overall percentage of correct classifications among all predictions made.
While its simplicity is appealing, accuracy might sometimes be deceptive,
especially in situations with skewed class distributions. Our model showcases
an accuracy improvement of $\sim$10% compared to pure image-based CNN model.
As we delve deeper, we encounter precision, which allows us to zoom into the
model’s positive predictions. Also known as the positive predictive value,
precision quantifies the fraction of true positive predictions amidst all
positive predictions made, a critical indicator when the implications of false
positives are substantial. In this regard, our model surpasses CNN with a
precision advantage of $\sim$16%. Our exploration then pivots towards recall
or sensitivity, another perspective-shifting metric that focuses on the actual
positive cases, computing the proportion that the model correctly identifies.
Its criticality surges when the repercussions of false negatives are high,
ensuring that the model captures all relevant instances. Our model presents a
recall improvement of $\sim$6.6% over the alternative CNN model. Bridging
precision and recall, we have the F1 score. This reconciling metric provides a
balanced measure of a model’s performance by combining both precision and
recall into a single entity. With its value oscillating between 0 (worst) and
1 (perfect precision and recall), the F1 score offers a comprehensive picture
of the model’s performance. Herein, our model boasts an F1 score elevation of
$\sim$10% in relation to the CNN model. Finally, we engage with the Area Under
the Receiver Operating Characteristics (AUROC), a metric that transcends
individual outcomes to evaluate overall model performance. It represents the
probability that the model will rank a randomly chosen positive instance
higher than a negative one. The true value of AUROC shines as it evaluates
both true positive and false positive rates, offering an all-encompassing
performance view across all classification thresholds. Within this sphere, our
model achieves an AUROC enhancement of $\sim$10% compared to its counterpart.
By analyzing these interconnected metrics, we elucidate the comprehensive
performance profile of our models. Examining Figure 6(a-e), the superior
performance of our MIML model is evident across all evaluation categories.
This clear edge substantiates the importance of incorporating both image and
cellular mechanical properties as inputs, demonstrating the efficacy of our
approach. Notably, the significant leap in performance introduced by our MIML
model suggests its effectiveness, underscoring a promising advancement in cell
classification methodologies.

To provide a more nuanced evaluation of our MIML model, we have devised a
composite visualization combining a bar chart Figure 6(f) and a scatter plot.
The bar chart portrays the mean accuracies for the training, testing, and
validation datasets derived from five-fold cross-validation, with each bar’s
height representing the mean accuracy and the attached error bars denoting the
variability in the results. Superimposed on this bar chart, we have a scatter
plot that displays the individual accuracies from each of the five cross-
validation trials. This layered presentation affords a more comprehensive
overview of the model’s performance. The close agreement in the training
accuracies across cross-validation trials, with an average increase of $\sim
10.4\%$ compared to the pure image-based CNN model, reinforces the model’s
reproducibility with the training dataset. Simultaneously, the proximity of
the validation and testing accuracies to each other, with an average
improvement of $\sim 10.5\%$ signifies the model’s robust generalizability,
suggesting the absence of overfitting. This combined visualization not only
confirms the reliability of our MIML model but also offers valuable insights
into its performance dynamics.

We investigate the diversity of the patterns presented in our data set. To do
so, we randomly selected three patches within the latent space and visualized
the images associated with them (Figure7(i-viii)). Interestingly, despite an
initial impression of high similarity among the images, the model was able to
differentiate between them. On a superficial level, all three groups appeared
quite alike, with subtle variations only perceptible upon meticulous
examination. However, when processed through our model, these seemingly subtle
differences were amplified, and the model distinctly categorized each group.
Group 2, while almost indistinguishable from the other groups by eye, was
identified by the model as possessing specific attributes that set it apart.
Similarly, Group 1, and Group 3, despite their visible similarities to each
other, were distinctly classified based on the model’s analysis of the
underlying patterns and structures in the data. This reveals the power and
sensitivity of our model in distinguishing between seemingly identical data
points. Even when human observers might struggle to discern any differences
due to their apparent similarities, the model is capable of picking up on
minute differences and categorizing the data accurately. This affirms the
strength of our model in dealing with complex, high-dimensional data and
underscores its potential utility in various fields where subtle variations in
data could hold significant implications.

##  3 Methods

###  3.1 Preparation of the microfluidic device

The microfluidic channels are produced via the conventional UV lithography
technique. Initially, channel designs are drafted using AutoCAD. Using the
direct laser writing tool, DWL 66+ (sourced from the Quattrone Nanofabrication
Facility at the Singh Center for Nanotechnology, University of Pennsylvania),
chrome masks are created. These masks then facilitate the creation of the
master pattern on an SU-8 2007 (MicroChem) layer on a silicon wafer, executed
at the Center for Photonics and Nanoelectronics (CPN) at Lehigh University.
The SU8-2007 is applied to the silicon wafer at a speed of 1000 rpm. Following
a soft bake phase, the SU-8 undergoes UV exposure using the Suss MA6/BA6.
Post-development, the SU-8 designs undergo a hard bake at 150°C for 30
minutes. Sylgard 184 PDMS, combined with its curing agent at a 10:1 ratio, is
poured onto the photoresist master. After a 2-hour degassing period for the
PDMS, it is allowed to cure overnight in an oven. Finally, the inlets and
outlets are created in the PDMS channel prior to its attachment to a large
coverslip, secured with oxygen plasma treatment.

###  3.2 Cell culture and data collection

The Human Colorectal Cancer cell line (HCT116), which was purchased from
American Type Culture Collection (ATCC), and cultured in Dulbecco’s Modified
Eagle’s Medium (DMEM, Gibco). The DMEM was meticulously supplemented with 10%
Fetal Bovine Serum (FBS, Gibco), and 100 U/mL of Penicillin Streptomycin (R&D
system. To maintain the cells in an optimal state, we change the culture
medium every other day. To passage or isolate to single cell, we employed a
0.05% Trypsin-EDTA (Bio-Techne Corporation. The Human Peripheral Blood
Mononuclear cells (PBMCs) were obtained from the Human Immunology Core at Penn
Medicine. These PBMCs were maintained in RPMI-1640 medium (ATCC), fortified
with 10% FBS and 100 U/mL of Penicillin-Streptomycin. All the bright field
images were captured using Nikon Eclipse TE2000S inverted microscope with a
Ximea CCD Camera. All the images were taken with the same setting for
comparison.

###  3.3 Feature extraction

During our investigation, we implemented an experimental protocol that guided
the cells through carefully constructed, narrow passages. Our ability to
detect and scrutinize these cells was enhanced by a straightforward yet potent
machine learning model - Yolov5. This model generated bounding boxes, a
crucial tool that helped us concentrate on each individual cell, even when
they were under the pressure of induced deformation. Within these defined
areas, we initiated a detailed analysis of several parameters. As a first
step, we computed the deformation index, a metric indicating the degree of
cellular deformation. Concurrently, we methodically measured the time taken by
each cell to travel the entire length of the channel, a metric defined as
Transition Time 1(c). We also evaluated the maximum velocity achieved by each
cell during its passage through the channel. Upon calculating the
aforementioned parameters, we segmented the cell image and stored it along
with its measured property for subsequent training of our MIML model.

##  4 Conclusion

In this study, we introduce a novel machine learning architecture, which we
refer to as MIML, specifically designed to integrate label-free cell imagery
with cell biomechanical property data. This innovative combination harnesses
the full potential of the morphological information intrinsic to each cell,
which is often overlooked or discarded in conventional machine learning
approaches. Our primary objective with MIML is to enhance the precision of
cell classification by broadening the range of cell characteristics
considered. The MLML architecture synergistically combines these two data
types to create a comprehensive cell profile. By integrating both label-free
images and biomechanical data, our model provides a more holistic
understanding of the cell. It can capture and utilize the rich morphological
information usually discarded by traditional machine learning models. As a
result, we have achieved a higher accuracy rate in cell classification
compared to models that only consider one type of data. Furthermore, this
methodology bears extensive implications across disciplines such as disease
diagnostics and the deciphering of cell behavior, laying the groundwork for
the creation of enhanced therapeutic tactics and treatment regimes. This
model’s application has been demonstrated in a use-case scenario involving the
classification of White Blood Cells (WBCs) and HCT116 cells. Nevertheless, the
inherent versatility of the model, facilitated by transfer learning, allows
for its potential deployment in the categorization of any cellular species.
This is particularly effective when the cells are morphologically similar yet
possess distinct biomechanical properties, thus extending the model’s
applicability beyond its initial design.

## Conflicts of interest

The authors declare no conflict of interest.

## Acknowledgements

This work was supported by the National Institutes of Health (NIH) grant
number R21EB033102, the National Science Foundation (NSF) grant number OAC
2215789 and CBET2039310, and the Pennsylvania Infrastructure Technology
Alliance (PITA) to Y.L.

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See pages - of supplimentary.pdf


# Special values of $p$-adic $L$-functions and Iwasawa $\lambda$-invariants of
Dirichlet characters

Heiko Knospe  Technische Hochschule Köln
Fakultät für Informations-, Medien- und Elektrotechnik
Institute of Computer and Communication Technology
Betzdorfer Str. 2
50679 Köln
Germany<EMAIL_ADDRESS>
###### Abstract.

We study the Iwasawa $\lambda$-invariant of Dirichlet characters $\chi$ for
odd primes $p$. From the behaviour of the $p$-adic $L$-function at $s=0$,
$s=1$ and $s=1-p$, we derive easily computable novel criteria to distinguish
between the cases $\lambda=0$, $\lambda=1$, $\lambda=2$ and $\lambda\geqslant
3$. In particular, we look at the case where the $p$-adic $L$-function and the
Iwasawa power series have a trivial (exceptional) zero. We give a condition
for $\lambda_{p}(\chi)>1$ using the $p$-adic valuation of the Bernoulli number
$B_{p,\chi\omega^{-1}}$ or alternatively $L_{p}(1,\chi)$. We also provide a
similar condition for $\lambda_{p}(\chi)>2$. The formulas are used to obtain
numerical data on $\lambda$-invariants, where either a prime or a Dirichlet
character is fixed. The data is consistent with our earlier conjecture on the
distribution of $\lambda$-invariants. Now the case of a trivial zero is
included, for which the distribution of $\lambda$-values is shifted by $1$.

###### Key words and phrases:

Iwasawa theory, Dirichlet characters, $p$-adic $L$-functions, Iwasawa
$\lambda$-invariant, trivial zero, special values

###### 2010 Mathematics Subject Classification:

Primary 11R23; Secondary 11R42, 11S80, 11M41

##  1\. Introduction

Let $p$ be an odd prime and let $\chi$ be a Dirichlet character. The Kubota-
Leopoldt $L$-function $\mathbf{L}_{p}(\chi,s)$ is a meromorphic function of
the $p$-adic variable $s$, which interpolates values of $L(s,\chi)$ along the
critical strip of non-positive integers [9]. More precisely, one has the
formula

(1.1)
$\mathbf{L}_{p}(1-n,\chi)\;=\;-\big{(}1-\chi\omega^{-n}(p)p^{n-1}\big{)}\times\frac{B_{n,\chi\omega^{-n}}}{n}\quad\text{at
every $n\geqslant 1$.}$

$\omega$ denotes the Teichmüller character mod $p$ and $B_{n,\chi\omega^{-n}}$
is the $n$-th $\chi\omega^{-n}$-twisted Bernoulli number.

Now suppose that $\chi$ is a non-trivial character of the first kind, namely
that $\chi=\theta\omega^{i}$ where $p$ does not divide the conductor of
$\theta$ and $i\in\\{0,\dots,p-2\\}$. We will also set
$\mathcal{O}_{\chi}:=\mathbb{Z}_{p}[\text{\rm Im}(\chi)]$. Iwasawa associated
a unique power series
$\mathcal{F}_{\chi}(T)=\sum_{j=0}^{\infty}c_{j}T^{j}\in\mathcal{O}_{\chi}[\\![T]\\!]$
satisfying

(1.2)
$\mathcal{F}_{\chi}\big{(}(1+p)^{s}-1\big{)}=\;\mathbf{L}_{p}(s,\chi)\quad\text{for
all $s\in\widehat{\overline{\mathbb{Q}}}_{p}$ with
$\big{|}s\big{|}_{p}\\!<p^{\frac{p-2}{p-1}}$.}$

The $\mu$-invariant of $\mathcal{F}_{\chi}$ vanishes by the Ferrero-Washington
theorem [6], and $\mathcal{F}_{\chi}$ factorises into a product of an
invertible power series and a uniquely determined distinguished polynomial.
The degree $\lambda=\lambda_{p}(\chi)$ equals the number of zeroes of
$\mathcal{F}_{\chi}(T)$ on the open unit disk.

In our article [1], we generalized a conjecture of Ellenberg-Jain-Venkatesh
[4] and predicted that the probability that $\lambda_{p}(\chi)=r$ is
approximately equal to

(1.3) $p^{-fr}\times\;\prod_{t>r}\;\big{(}1-p^{-ft}\big{)},$

where $f\geqslant 1$ denotes the multiplicative order of the prime $p$ modulo
$\text{\rm ord}(\chi)$.

In the present article, we give new new formulas to compute the
$\lambda$-invariant. In particular, we look at the case where the $p$-adic
$L$-function and the associated power series have a trivial (also called
exceptional) zero at $s=0$ and $T=0$. We provide numerical evidence that our
prediction on the distribution of $\lambda$-invariants remains true in that
case if we shift $\lambda$ by $1$.

The following Proposition follows easily from the definition of the
$\lambda$-invariant.

######  Proposition 1.1.

Let $K/\mathbb{Q}_{p}$ be a finite extension with ring of integers
$\mathcal{O}$ and let $v_{p}$ be the normalized valuation of $K$ where
$v_{p}(p)=1$. Let $\mathcal{F}(T)\in\mathcal{O}[[T]]$ and
$t_{0}\in\mathcal{O}$ such that $v(t_{0})>0$. Let
$p_{n}(T)=a_{0}+a_{1}(T-t_{0})+\dots+a_{n}(T-t_{0})^{n}$ be the associated
$n$-th Taylor polynomial at $T=t_{0}$. Choose any $b\in\mathcal{O}$ such that
$v_{p}(b-t_{0})=1$. Then $\lambda_{p}(\mathcal{F})>0$ iff
$v_{p}(\mathcal{F}(t_{0}))>0$. For any integer $n>0$ one has

$\lambda_{p}(\mathcal{F})>n\Longleftrightarrow\lambda_{p}(\mathcal{F})>n-1\text{
and }v_{p}(\mathcal{F}(b)-p_{n-1}(b))>n.$

The latter condition can be checked modulo $p^{n+1}$ (or any exponent $>n$),
i.e., it suffices to compute $\mathcal{F}(b)$ and $p_{n-1}(b)$ up to precision
$O(p^{n+1})$.

###### Proof.

The $\lambda$-invariant of $\mathcal{F}(T)$ can be computed using an expansion
around any $t_{0}\in\mathcal{O}$ with $v(t_{0})>0$. The first part of the
assertion is obvious and

$\mathcal{F}(b)-p_{n-1}(b)=a_{n}(b-t_{0})^{n}+a_{n+1}(b-t_{0})^{n+1}+\dots$

Assuming that $\lambda_{p}(\mathcal{F})>n-1$, one has
$\lambda_{p}(\mathcal{F})>n$ iff $v_{p}(a_{n})>0$. We note that
$v_{p}((b-t_{0})^{n})=n$ and $v_{p}(a_{n+1}(b-t_{0})^{n+1}+\dots)\geqslant
n+1$. Hence $v_{p}(a_{n})>0$ iff
$v_{p}(a_{n}(b-t_{0})^{n}+a_{n+1}(b-t_{0})^{n+1}+\dots)>n$. This can be
checked modulo $p^{n+1}$. ∎

######  Corollary 1.2.

Let $\mathcal{F}_{\chi}(T)$ be the Iwasawa power series associated to $\chi$,
and let $t_{0},\ b\in\mathcal{O}_{\chi}$ as in Proposition 1.1. Then:

  1. (i)

$\lambda_{p}(\chi)>0\Longleftrightarrow v_{p}(\mathcal{F}_{\chi}(t_{0}))>0$.

  2. (ii)

$\lambda_{p}(\chi)>1\Longleftrightarrow$ (i) and
$v_{p}(\mathcal{F}_{\chi}(b)-\mathcal{F}_{\chi}(t_{0}))>1$.

  3. (iii)

$\lambda_{p}(\chi)>2\Longleftrightarrow$ (i) and (ii) and
$v_{p}(\mathcal{F}_{\chi}(b)-\mathcal{F}_{\chi}^{\prime}(t_{0})(b-t_{0})-\mathcal{F}_{\chi}(t_{0}))>2$.

(i) can be checked modulo $p$, (ii) modulo $p^{2}$ and (iii) modulo $p^{3}$.

We use the above results to compute $\lambda$-invariants of Dirichlet
characters. For the numerical computations, we propose two alternative sets of
parameters:

1. $t_{0}=0$ and $b=(1+p)^{1-p}-1$. Then $v_{p}(b-t_{0})=v_{p}(b)=1$ and the special values are

(1.4) $\displaystyle\mathcal{F}_{\chi}(0)$
$\displaystyle=L_{p}(0,\chi)=-(1-\chi\omega^{-1}(p))B_{1,\chi\omega^{-1}},$
(1.5) $\displaystyle\mathcal{F}_{\chi}^{\prime}(0)$
$\displaystyle=\frac{L_{p}^{\prime}(0,\chi)}{\log_{p}(1+p)},$ (1.6)
$\displaystyle\mathcal{F}_{\chi}(b)$
$\displaystyle=L_{p}(1-p,\chi)=-(1-\chi\omega^{-1}(p)p^{p-1})\frac{B_{p,\chi\omega^{-1}}}{p}\
.$

2. $t_{0}=p$ and $b=0$. Then $v_{p}(b-t_{0})=v_{p}(-p)=1$ and

(1.7) $\displaystyle\mathcal{F}_{\chi}(p)$ $\displaystyle=L_{p}(1,\chi),$
(1.8) $\displaystyle\mathcal{F}_{\chi}^{\prime}(p)$
$\displaystyle=\frac{L_{p}^{\prime}(1,\chi)}{(1+p)\log_{p}(1+p)}.$

We need explicit formulas for $L_{p}^{\prime}(0,\chi)$, $L_{p}(1,\chi)$ and
$L_{p}^{\prime}(1,\chi)$. For the derivative of $L_{p}(s,\chi)$ at $s=0$ we
recall a result of Ferrero-Greenberg:

######  Proposition 1.3.

([5] Prop. 1) Let $d$ be the conductor of $\chi\omega^{-1}$. If $p\nmid d$
then

$L_{p}^{\prime}(0,\chi)=\left(\sum_{a=1}^{d}\chi\omega^{-1}(p)\log_{p}\Gamma_{p}\left(\frac{a}{d}\right)\right)+(1-\chi\omega^{-1}(p))B_{1,\chi\omega^{-1}}\log_{p}(d),$

where $\Gamma_{p}$ is the $p$-adic gamma function.

We will also need a formula when $p$ divides the conductor of
$\chi\omega^{-1}$. For this we refer to a result of Washington [12]:

######  Proposition 1.4.

Let $F$ be any multiple of $p$ and the conductor of $\chi$. Then:

$L_{p}^{\prime}(\chi,0)=\left(\sum_{\begin{subarray}{c}a=1\\\ p\nmid
a\end{subarray}}^{F}\chi\omega^{-1}(a)G_{p}\left(\frac{a}{F}\right)\right)+(1-\chi\omega^{-1}(p))B_{1,\chi\omega^{-1}}\log_{p}(F)$

where $G_{p}$ is Diamond’s $p$-adic log gamma function defined by

$G_{p}(X)=\left(X-\frac{1}{2}\right)\log_{p}X-X+\sum_{j=2}^{\infty}\frac{B_{j}}{j(j-1)}X^{1-j}$

for $|X|_{p}>1$ (see [2]).

For $s=1$ we obtain the following formulas:

######  Proposition 1.5.

Let $\chi=\theta\omega^{i}$ and let $d$ be the conductor of $\theta$, where
$p\nmid d$. Set $F=pd$. Then we have

$\displaystyle L_{p}(1,\chi)$
$\displaystyle\equiv-\sum_{\begin{subarray}{c}a=1\\\ p\nmid
a\end{subarray}}^{F}\chi(a)\left(\frac{1}{F}\log_{p}(a)-\frac{1}{2a}-\frac{F}{12a^{2}}\right)\mod
p^{3},$ $\displaystyle L_{p}^{\prime}(1,\chi)$
$\displaystyle\equiv\sum_{\begin{subarray}{c}a=1\\\ p\nmid
a\end{subarray}}^{F}\chi(a)\left(\frac{F}{12a^{2}}-\frac{\log_{p}(a)}{2a}-\frac{\log_{p}(a)F}{12a^{2}}+\frac{\log_{p}(a)^{2}}{2F}\right)\mod
p^{3}.$

###### Proof.

We look at the expansion
$L_{p}(s,\chi)=a_{0}+a_{1}(s-1)+a_{2}(s-1)^{2}+\dots$. Washington shows
$|a_{0}|<1$ and $a_{i}\equiv 0\mod p$ for all $i\geqslant 1$ (see [13] Theorem
5.12). We proceed similarly as in the proof of ibid., but calculate
$a_{0}=L_{p}(1,\chi)$ and $a_{1}=L_{p}^{\prime}(1,\chi)$ modulo $p^{3}$
instead of modulo $p$. We skip the details. ∎

Furthermore, the above expansion
$L_{p}(s,\chi)=a_{0}+a_{1}(s-1)+a_{2}(s-1)^{2}+\dots$ and $p\mid a_{i}$ for
$i\geqslant 1$ yields the congruences

(1.9) $\displaystyle L_{p}(1-p,\chi)$
$\displaystyle=a_{0}+a_{1}(-p)+a_{2}(-p)^{2}+\dots\equiv
a_{0}=L_{p}(1,\chi)\mod p^{2},$ (1.10) $\displaystyle
L_{p}^{\prime}(1-p,\chi)$
$\displaystyle=a_{1}+a_{2}(-p)+a_{3}(-p)^{2}+\dots\equiv
a_{1}=L_{p}^{\prime}(1,\chi)\mod p^{2}.$

##  2\. The Trivial Zero Case

It is well known that the interpolation factor, i.e., the Euler factor at $p$
for $L(s,\chi\omega^{-n})$, can produce a trivial zero of $L_{p}(s,\chi)$ at
$s=0$. Ferrero and Greenberg have shown in [5] that for Dirichlet characters
$\chi$, $L_{p}(s,\chi)$ can have at most a simple zero at $s=0$.

######  Proposition 2.1.

If $\chi\omega^{-1}(p)=1$ then $\mathbf{L}_{p}(s,\chi)$ has a trivial zero at
$s=0$, in which case $\mathcal{F}_{\chi}(T)$ also has a trivial zero at $T=0$.
This happens precisely if $\chi=\theta\omega$, where $\theta$ is odd of
conductor prime to $p$, and $\theta(p)=\chi\omega^{-1}(p)=1$. In this
situation we have $\lambda_{p}(\chi)\geqslant 1$.

###### Proof.

This follows from the interpolation property (1.1) for $n=1$. Note that
$\theta\omega^{i}(p)=0$ if $i\not\equiv 0\mod p-1$. ∎

######  Example.

Let $\chi$ be the non-trivial character associated to a quadratic extension
$K/\mathbb{Q}$. Then $L_{p}(s,\chi\omega)$ has a trivial zero at $s=0$ iff
$\chi$ is odd and $\chi(p)=1$. This is equivalent to $K$ being imaginary
quadratic and $p$ being a prime which splits in $K$.

We now give formulas to distinguish the cases $\lambda_{p}(\chi)=1$,
$\lambda_{p}(\chi)=2$ and $\lambda_{p}(\chi)\geqslant 3$.

######  Theorem 2.2.

Suppose $\chi\omega^{-1}(p)=1$ so that $L_{p}(s,\chi)$ has a trivial zero at
$s=0$.

  1. (i)

One has $\lambda_{p}(\chi)>1$ if and only if

$v_{p}(B_{p,\chi\omega^{-1}})>2.$

An equivalent condition is that

$v_{p}(L_{p}(1,\chi))>1.$

  2. (ii)

Assume that $\lambda_{p}(\chi)>1$. Then $\lambda_{p}(\chi)>2$ if and only if

$v_{p}\left(-(1-p^{p-1})\frac{B_{p,\chi\omega^{-1}}}{p}-\frac{L_{p}^{\prime}(0,\chi)}{\log_{p}(1+p)}((1+p)^{1-p}-1)\right)>2.$

An equivalent condition is that

$v_{p}\left(\frac{L_{p}^{\prime}(1,\chi)}{(1+p)\log_{p}(1+p)}\,p-L_{p}(1,\chi)\right)>2.$

###### Proof.

We choose $b=(1+p)^{1-p}-1$. Then $v_{p}(b)=1$, and by (1.6),
$\mathcal{F}_{\chi}(b)=(1-\chi\omega^{-1}(p)p^{p-1})\frac{B_{p,\chi\omega^{-1}}}{p}=(1-p^{p-1})\frac{B_{p,\chi\omega^{-1}}}{p}$.
From (1.9) we obtain $\mathcal{F}_{\chi}(b)\equiv L_{p}(1,\chi)\mod p^{2}$.
Now part (i) of the assertion follows from Proposition 1.2 (ii), and part (ii)
follows from Corollary 1.2 (iii), (1.5) and (1.8). ∎

######  Remark.

All of the above special values can be efficiently computed. Let $d$ be the
conductor of $\chi\omega^{-1}$. By Proposition 1.3, we have

$L_{p}^{\prime}(0,\chi)=\sum_{a=1}^{d}\chi\omega^{-1}(p)\log_{p}\Gamma_{p}\left(\frac{a}{d}\right)$

in the trivial zero case when $p\nmid d$. Formulas for $L_{p}(1,\chi)$ and
$L_{p}^{\prime}(1,\chi)$ are given in Proposition 1.5.

######  Corollary 2.3.

Let $\chi$ be a Dirichlet character satisfying $\chi\omega^{-1}(p)=1$. Let $K$
be the abelian number field associated to the odd character $\chi\omega^{-1}$.
The field $K$ is totally imaginary and $[K:\mathbb{Q}]=\text{\rm
ord}(\chi)=2r_{2}$. Then the p-adic $L$-functions
$L_{p}((\chi\omega^{-1})^{i}\omega,s)=L_{p}(\chi^{i}\omega^{1-i},s)$ of $K$,
$i=1,3,\dots,2r_{2}-1$, all vanish at $s=0$.

###### Proof.

The $p$-adic $L$-functions of $K$ are $L_{p}(s,\chi^{i}\omega^{1-i})$ with
$i=1,3,\dots,2r_{2}-1$. Since $(\chi^{i}\omega^{-i})(p)=1$ these $p$-adic
$L$-functions have a trivial zero at $s=0$. ∎

The above Corollary implies $\lambda(K)\geqslant r_{2}$ for the field $K$
associated to $\chi\omega^{-1}$. This is in fact known for more general number
fields and $\mathbb{Z}_{p}$-extensions:

######  Proposition 2.4.

Let $K$ be a finite extension of $\mathbb{Q}$ and assume that $p$ splits
completely in $K/\mathbb{Q}$. Let $r_{2}$ be the number of pairs of complex
embeddings of $K$. Assume that $K_{\infty}$ is a $\mathbb{Z}_{p}$-extension of
$K$ such that every prime above $p$ is ramified in $K_{\infty}/K$, e.g., the
cyclotomic extension of $K$. Then the $\lambda$-invariant of $K$ (defined by
the growth of the $p$-class groups in the $\mathbb{Z}_{p}$-extension of $K$)
satisfies

$\lambda_{p}(K)\geqslant r_{2}.$

###### Proof.

This result is known (see [10], [7]) and can be shown using only Galois theory
and class field theory. Let
$\Gamma=\text{Gal}(K_{\infty}/K)\cong\mathbb{Z}_{p}$ and let $\widetilde{K}$
be the compositum of all $\mathbb{Z}_{p}$-extensions of $K$. Then
$G:=\mathrm{Gal}(\widetilde{K}/K)\cong\mathbb{Z}_{p}^{r_{2}+1+\delta}$ with
$\delta\geqslant 0$ (see [13] Theorem 13.4) and
$X:=\mathrm{Gal}(\widetilde{K}/K_{\infty})\cong\mathbb{Z}_{p}^{r_{2}}$. We
have an exact sequence

$0\longrightarrow X\rightarrow G\longrightarrow\Gamma\longrightarrow 0.$

Let $\mathfrak{p}$ be a prime of $K$ above $p$, and let $\mathfrak{P}$ be a
prime of $\widetilde{K}$ that lies above $\mathfrak{p}$. We define
$T=T(\mathfrak{P}/\mathfrak{p})\subset G$ to be the associated inertia
subgroup. We claim that projection of $T$ to $\Gamma$ is injective. To this
end, we use the assumption that $\chi\omega^{-1}(p)=1$, which implies that the
primes $\mathfrak{p}$ above $p$ are fully split in $K$. Hence the local unit
group $U_{\mathfrak{p}}$ is isomorphic to $\mathbb{Z}_{p}^{\times}$, and the
pro-$p$-part $U^{0}_{\mathfrak{p}}$ is isomorphic to $\mathbb{Z}_{p}$. Class
field theory then implies that the inertia group $T$ is the image of
$U^{0}_{\mathfrak{p}}$ under the Artin map. Since $\mathfrak{p}$ is ramified
in the extension $K_{\infty}/K$, the inertia group $T$ cannot be trivial and
must be isomorphic to $\mathbb{Z}_{p}$. Hence the reduction map from $T$ to
$\Gamma$ is injective. Therefore, $T\cap X=0$ which shows that the primes
above $p$ are unramified in the extension $\widetilde{K}/K_{\infty}$. We thus
obtain that $\widetilde{K}/K_{\infty}$ is unramified everywhere. The Galois
group of this unramified extension is isomorphic to $\mathbb{Z}_{p}^{r_{2}}$,
and the $\lambda_{p}$-invariant of $K$ must be $\geqslant r_{2}$. ∎

######  Example.

Let $p\equiv 1\mod 4$ and let $\chi=\chi_{4}\omega$, where $\chi_{4}$ denotes
the odd Dirichlet character of conductor $4$ associated to the imaginary
quadratic field $\mathbb{Q}(i)$. Then $\chi_{4}(p)=1$ and the $p$-adic
$L$-function $L_{p}(s,\chi)$ has a trivial zero at $s=0$. Let $E_{n}$ be the
$n$-th Euler number defined by

$\sum_{n=0}^{\infty}E_{n}\frac{x^{n}}{n!}=\frac{2}{e^{x}+e^{-x}}.$

On the other hand, the generating function for $B_{n,\chi_{4}}$ is given by

$\frac{xe^{x}-xe^{3x}}{e^{4x}-1}=x\frac{e^{x}(1-e^{2x})}{(e^{2x}+1)(e^{2x}-1)}=-x\frac{1}{e^{x}+e^{-x}}.$

Hence we get $\frac{E_{n-1}}{2}=-\frac{B_{n,\chi_{4}}}{n}$. By Theorem 2.2
(i), $\lambda_{p}(\chi_{4})=\lambda_{p}(\mathbb{Q}(i))>1$ iff
$v_{p}(B_{p,\chi_{4}})>2\Longleftrightarrow v_{p}(E_{p-1})>1$.

A similar statement holds true for the odd Dirichlet character $\chi_{3}$ of
order $3$. Now let $p\equiv 1\mod 3$ and $\chi=\chi_{3}\omega$. Here we
consider the $n$-th Glaisher I-number $G_{n}$ defined by

$\sum_{n=0}^{\infty}G_{n}\frac{x^{n}}{n!}=\frac{3/2}{1+e^{x}+e^{-x}}.$

The generating function for $B_{n,\chi_{3}}$ is given by

$\frac{xe^{x}-xe^{2x}}{e^{3x}-1}=x\frac{e^{x}(1-e^{x})}{(e^{x}-1)(e^{2x}+e^{x}+1)}=-x\frac{1}{1+e^{x}+e^{-x}}.$

We obtain $\frac{2}{3}G_{n-1}=-\frac{B_{n,\chi_{3}}}{n}$ and
$\lambda_{p}(\chi_{3})=\lambda_{p}(\mathbb{Q}(\sqrt{-3}))>1$ iff
$v_{p}(G_{p-1})>1$. These results for Euler and Glaisher numbers were also
established by other methods in [11].

##  3\. The General Case

We now consider the general situation when no trivial zero is present. The
following theorem provides criteria for the cases $\lambda_{p}(\chi)=0$,
$\lambda_{p}(\chi)=1$, $\lambda_{p}(\chi)=2$ and $\lambda_{p}(\chi)\geqslant
3$. Of course, part (i) of the following theorem is well known.

######  Theorem 3.1.

Suppose $\chi\omega^{-1}(p)\neq 1$. Then:

  1. (i)

$\lambda_{p}(\chi)>0$ if and only if
$v_{p}(B_{1,\chi\omega^{-1}})=v_{p}(L_{p}(0,\chi))>0$.

  2. (ii)

Assume that $\lambda_{p}(\chi)>0$. Then $\lambda_{p}(\chi)>1$ if and only if

$v_{p}\left(-(1-\chi\omega^{-1}(p)p^{p-1})\frac{B_{p,\chi\omega^{-1}}}{p}+(1-\chi\omega^{-1}(p))B_{1,\chi\omega^{-1}}\right)>1.$

An equivalent condition is

$v_{p}(-(1-\chi\omega^{-1}(p))B_{1,\chi\omega^{-1}}-L_{p}(1,\chi))>1.$

  3. (iii)

Assume that $\lambda_{p}(\chi)>1$. Then $\lambda_{p}(\chi)>2$ if and only if

$\displaystyle
v_{p}\left(-(1-\chi\omega^{-1}(p)p^{p-1})\frac{B_{p,\chi\omega^{-1}}}{p}+(1-\chi\omega^{-1}(p))B_{1,\chi\omega^{-1}}\right.$
$\displaystyle-\frac{L_{p}^{\prime}(0,\chi)}{\log_{p}(1+p)}((1+p)^{1-p}-1)\bigg{)}>2.$

An equivalent condition is that

$v_{p}\left(-(1-\chi\omega^{-1}(p))B_{1,\chi\omega^{-1}}+\frac{L_{p}^{\prime}(1,\chi)}{(1+p)\log_{p}(1+p)}\,p-L_{p}(1,\chi)\right)>2.$

###### Proof.

(ii) and (iii) follow from Corollary 1.2 (ii), (iii) and equations (1.4),
(1.5), (1.6), (1.7), (1.8). ∎

######  Remark.

All of the above special values can be efficiently computed using Propositions
1.3, 1.4 und 1.5.

######  Remark.

Let us fix an odd character $\theta$, and let $K$ be the associated totally
imaginary field of degree $2r_{2}$. We know from Section 2 that
$\lambda_{p}(\theta\omega)\geqslant 1$ for all primes $p$ which are split in
$K/\mathbb{Q}$, i.e. primes $p$ with $\theta(p)=1$. The number of such primes
is infinite. We can then use the $p$-adic valuation of $B_{p,\theta}$ or the
valuation of $L_{p}(1,\chi)$ to determine whether
$\lambda_{p}(\theta\omega)>1$ (see Theorem 2.2 (i)).

On the other hand, consider the non-split unramified primes $p$, i.e. primes
with $\theta(p)\neq 1$. There are infinitely many such primes, but only
finitely many have the property that $\lambda_{p}(\theta\omega)>0$ since
$v_{p}(B_{1,\theta})=0$ for almost all primes $p$.

Furthermore, there are corresponding statements for the cyclotomic
$\lambda$-invariants of $K$. By considering all odd powers $\theta$,
$\theta^{3}$, $\dots$, $\theta^{\text{\rm ord}(\theta)-1}$, we have
$\lambda_{p}(K)\geqslant r_{2}$ for all split primes, and
$\lambda_{p}(K)>r_{2}$ iff $v_{p}(B_{p,\theta^{i}})>1$ for at least one odd
power $\theta^{i}$. However, among the non-split primes $p$, only finitely
many have the property that $\lambda_{p}(K)>0$.

##  4\. Numerical Data

We extend our work [1] where we predicted the distribution of
$\lambda$-invariants. Computations are much faster with the new formulas. We
also collect new data for $\lambda$-invariants in the trivial zero case
supporting our prediction. Note that we cannot distinguish between
$\lambda$-invariants $\geqslant 3$ with the new formulas. However, the case
$\lambda>3$ is rare with a predicted probability of less than $p^{-4}$.

In general, our computations use the special values of the first set of
parameters (see Section 1): $B_{1,\chi\omega^{-1}}$, $B_{p,\chi\omega^{-1}}$
and $L_{p}^{\prime}(0,\chi)$. In the trivial zero case,
$B_{1,\chi\omega^{-1}}$ is not needed. It is sufficient to compute these
special values mod $p$ (for $\lambda>0$), mod $p^{2}$ (for $\lambda>1$) or mod
$p^{3}$ (for $\lambda>2$). This speeds up the computations, in particular of
the $p$-adic log gamma function. We have sample checked the results by using
other formulas, in particular our second set of parameters, and also the
$p$-adic expansions in [8] and calculations by other authors, e.g. [3].

###  4.1. Distribution of $\lambda$-invariants in the trivial zero case

First, we provide numerical data for the trivial zero case and fixed primes.
In this situation, the predicted distribution of $\lambda$-values (see [1])
should be shifted by $1$.

Using the above formulas, we computed the $\lambda$-invariants of
$\chi=\theta\omega$, where $\theta$ is odd of conductor $<100,000$ and
$\theta(p)=1$. The first row gives the predicted distribution and the second
row the number $N$ of tested characters as well as the computed proportions.
For $p=3$ and $\text{\rm ord}(\chi)=2$ we still see a larger deviation from
the prediction (compare [1]), but the difference becomes smaller as we compute
$\lambda$-invariants of characters with larger conductor.

$p=3$, $ord(\chi)=2$
$N$ $\lambda=1$ $\lambda=2$ $\lambda\geqslant 3$ pred. $0.5601$ $0.2801$
$0.1596$ $11404$ $0.6121$ $0.2604$ $0.1275$

$p=3$, $ord(\chi)=4$
$N$ $\lambda=1$ $\lambda=2$ $\lambda\geqslant 3$ pred. $0.8766$ $0.1096$
$0.0139$ $17732$ $0.8901$ $0.0976$ $0.0123$

$p=5$, $ord(\chi)=2$
$N$ $\lambda=1$ $\lambda=2$ $\lambda\geqslant 3$ pred. $0.7603$ $0.1901$
$0.0496$ $12667$ $0.7782$ $0.1768$ $0.0450$

$p=5$, $ord(\chi)=4$
$N$ $\lambda=1$ $\lambda=2$ $\lambda\geqslant 3$ pred. $0.7603$ $0.1901$
$0.0496$ $13680$ $0.7662$ $0.1834$ $0.0504$

$p=5$, $ord(\chi)=6$
$N$ $\lambda=1$ $\lambda=2$ $\lambda\geqslant 3$ pred. $0.9584$ $0.0399$
$0.0017$ $37292$ $0.9578$ $0.0407$ $0.0016$

$p=7$, $ord(\chi)=2$
$N$ $\lambda=1$ $\lambda=2$ $\lambda\geqslant 3$ pred. $0.8368$ $0.1395$
$0.0237$ $13299$ $0.8419$ $0.1341$ $0.0239$

$p=7$, $ord(\chi)=4$
$N$ $\lambda=1$ $\lambda=2$ $\lambda\geqslant 3$ pred. $0.9792$ $0.0204$
$0.0004$ $21244$ $0.9797$ $0.0197$ $0.0007$

$p=7$, $ord(\chi)=6$
$N$ $\lambda=1$ $\lambda=2$ $\lambda\geqslant 3$ pred. $0.8368$ $0.1395$
$0.0237$ $27056$ $0.8361$ $0.1419$ $0.0220$

$p=11$, $ord(\chi)=2$
$N$ $\lambda=1$ $\lambda=2$ $\lambda\geqslant 3$ pred. $0.9008$ $0.0901$
$0.0091$ $13931$ $0.9058$ $0.0847$ $0.0095$

$p=11$, $ord(\chi)=4$
$N$ $\lambda=1$ $\lambda=2$ $\lambda\geqslant 3$ pred. $0.9917$ $0.0083$
$0.0001$ $23252$ $0.9895$ $0.0104$ $0.0001$

$p=11$, $ord(\chi)=6$
$N$ $\lambda=1$ $\lambda=2$ $\lambda\geqslant 3$ pred. $0.9917$ $0.0083$
$0.0001$ $41478$ $0.9920$ $0.0079$ $0.0002$

$p=13$, $ord(\chi)=2$
$N$ $\lambda=1$ $\lambda=2$ $\lambda\geqslant 3$ pred. $0.9172$ $0.0764$
$0.0064$ $14110$ $0.9199$ $0.0728$ $0.0073$

$p=13$, $ord(\chi)=4$
$N$ $\lambda=1$ $\lambda=2$ $\lambda\geqslant 3$ pred. $0.9172$ $0.0764$
$0.0064$ $18806$ $0.9210$ $0.0726$ $0.0064$

$p=13$, $ord(\chi)=6$
$N$ $\lambda=1$ $\lambda=2$ $\lambda\geqslant 3$ pred. $0.9172$ $0.0764$
$0.0064$ $36440$ $0.9189$ $0.0743$ $0.0068$

$p=17$, $ord(\chi)=2$
$N$ $\lambda=1$ $\lambda=2$ $\lambda\geqslant 3$ pred. $0.9377$ $0.0586$
$0.0037$ $14352$ $0.9392$ $0.0575$ $0.0033$

$p=19$, $ord(\chi)=2$
$N$ $\lambda=1$ $\lambda=2$ $\lambda\geqslant 3$ pred. $0.9446$ $0.0525$
$0.0029$ $14441$ $0.9463$ $0.0504$ $0.0033$

$p=23$, $ord(\chi)=2$
$N$ $\lambda=1$ $\lambda=2$ $\lambda\geqslant 3$ pred. $0.9546$ $0.0434$
$0.0020$ $14569$ $0.9552$ $0.0428$ $0.0020$

$p=29$, $ord(\chi)=2$
$N$ $\lambda=1$ $\lambda=2$ $\lambda\geqslant 3$ pred. $0.9643$ $0.0344$
$0.0012$ $14695$ $0.9666$ $0.0324$ $0.0010$

$p=31$, $ord(\chi)=2$
$N$ $\lambda=1$ $\lambda=2$ $\lambda\geqslant 3$ pred. $0.9667$ $0.0322$
$0.0011$ $14728$ $0.9659$ $0.0328$ $0.0013$

$p=31$, $ord(\chi)=6$
$N$ $\lambda=1$ $\lambda=2$ $\lambda\geqslant 3$ pred. $0.9667$ $0.0322$
$0.0011$ $41770$ $0.9672$ $0.0318$ $0.0011$

$p=37$, $ord(\chi)=2$
$N$ $\lambda=1$ $\lambda=2$ $\lambda\geqslant 3$ pred. $0.9722$ $0.0270$
$0.0008$ $14789$ $0.9734$ $0.0260$ $0.0007$

$p=41$, $ord(\chi)=2$
$N$ $\lambda=1$ $\lambda=2$ $\lambda\geqslant 3$ pred. $0.9750$ $0.0244$
$0.0006$ $14841$ $0.9749$ $0.0245$ $0.0007$

$p=43$, $ord(\chi)=2$
$N$ $\lambda=1$ $\lambda=2$ $\lambda\geqslant 3$ pred. $0.9762$ $0.0232$
$0.0006$ $14856$ $0.9768$ $0.0228$ $0.0004$

$p=43$, $ord(\chi)=6$
$N$ $\lambda=1$ $\lambda=2$ $\lambda\geqslant 3$ pred. $0.9762$ $0.0232$
$0.0006$ $46382$ $0.9777$ $0.0218$ $0.0005$

$p=47$, $ord(\chi)=2$
$N$ $\lambda=1$ $\lambda=2$ $\lambda\geqslant 3$ pred. $0.9783$ $0.0213$
$0.0005$ $14884$ $0.9789$ $0.0204$ $0.0007$

$p=97$, $ord(\chi)=2$
$N$ $\lambda=1$ $\lambda=2$ $\lambda\geqslant 3$ pred. $0.9896$ $0.0103$
$0.0001$ $15040$ $0.9897$ $0.0102$ $0.0001$

###  4.2. Primes with $\lambda>1$ in the trivial zero case

We fixed an odd character $\theta$ and used Theorem 2.2 (i) to compute all
small primes $p$ such that $L_{p}(s,\theta\omega)$ has a trivial zero and
$\lambda_{p}(\theta\omega)>1$. This was also done by Dummit et al. [3] for
imaginary quadratic characters using other methods. Below, we list the
combinations of characters $\chi=\theta\omega$ of order $<10$, conductor
$<100$ and primes $p<1000$ having a trivial zero and satisfying
$\lambda_{p}(\chi)>1$. For a fixed character, we expect (see [3, 1]) that the
number of such primes with $p\leqslant X$ is $O(\log(\log(X))$. Furthermore,
almost all of these primes should have residue class degree $f=1$, i.e. only a
finite number is expected to have degree $f\geqslant 2$.

order | conductor | field | primes
---|---|---|---
$2$ | $3$ | $\mathbb{Q}(\sqrt{-3})$ |  $13$, $181$
$2$ | $11$ | $\mathbb{Q}(\sqrt{-11})$ | $5$
$2$ | $19$ | $\mathbb{Q}(\sqrt{-19})$ | $11$
$2$ | $24$ | $\mathbb{Q}(\sqrt{-6})$ | $131$
$2$ | $31$ | $\mathbb{Q}(\sqrt{-31})$ | $227$
$2$ | $35$ | $\mathbb{Q}(\sqrt{-35})$ |  $3$, $13$
$2$ | $47$ | $\mathbb{Q}(\sqrt{-47})$ |  $3$, $17$, $157$
$2$ | $51$ | $\mathbb{Q}(\sqrt{-51})$ | $5$
$2$ | $52$ | $\mathbb{Q}(\sqrt{-13})$ | $113$
$2$ | $56$ | $\mathbb{Q}(\sqrt{-14})$ | $3$
$2$ | $71$ | $\mathbb{Q}(\sqrt{-71})$ | $29$
$2$ | $83$ | $\mathbb{Q}(\sqrt{-83})$ |  $17$, $41$
$2$ | $84$ | $\mathbb{Q}(\sqrt{-21})$ |  $107$, $173$
$2$ | $88$ | $\mathbb{Q}(\sqrt{-22})$ |  $23$, $29$

order | conductor | field (polynomial) | primes
---|---|---|---
$4$ | $16$ | $x^{4}+4x^{2}+2$ | $97$
$4$ | $29$ | $x^{4}+x^{3}+4x^{2}+20x+23$ | $181$
$6$ | $9$ | $x^{6}+x^{3}+1$ | $19$
$6$ | $19$ | $x^{6}+x^{5}+2x^{4}-8x^{3}-x^{2}+5x+7$ | $7$
$6$ | $28$ | $x^{6}+5x^{4}+6x^{2}+1$ | $337$
$6$ | $39$ | $x^{6}-x^{5}+5x^{4}+6x^{3}+15x^{2}+4x+1$ | $31$
$6$ | $52$ | $x^{6}+13x^{4}+26x^{2}+13$ | $31$
$6$ | $63$ | $x^{6}-28x^{3}+343$ | $193$
$6$ | $63$ | $x^{6}+35x^{3}+343$ | $241$
$6$ | $91$ | $x^{6}-x^{5}+21x^{4}-22x^{3}+58x^{2}+23x+155$ | $31$
$8$ | $85$ | $x^{8}-x^{7}+10x^{6}+6x^{5}+49x^{4}-129x^{3}+500x^{2}+2044x+1616$ | $433$
$8$ | $96$ | $x^{8}+24x^{6}+180x^{4}+432x^{2}+162$ | $17$

All primes listed above have residue class degree $f=1$, but we have also
found some examples (for characters of conductor $>100$) primes with $f=2$
(see below). It is not surprising that these primes are small.

order | conductor | field (polynomial) | primes
---|---|---|---
$4$ | $187$ | $x^{4}-x^{3}+45x^{2}+x+562$ | $19$
$6$ | $259$ | $x^{6}-x^{5}+22x^{4}+398x^{3}+1051x^{2}-x+23605$ | $5$
$8$ | $187$ | $x^{8}-x^{7}+44x^{6}-45x^{5}+423x^{4}-877x^{3}+1826x^{2}-3515x+4591$ | $13$

###  4.3. Distribution of $\lambda$-invariants in the general case

We also provide data for the general case (without a trivial zero), which
supplements the numerical data given in [1]. With the new formulas of this
article, we computed the $\lambda$-invariants of $\chi=\theta\omega$, where
$\theta$ is odd of conductor $<100,000$ and $\theta(p)\neq 1$. As in Section
4.1 above, the first row gives the predicted distribution and the second row
contains the number $N$ of tested characters as well as the computed
proportions.

$p=3$, $ord(\chi)=2$
$N$ $\lambda=0$ $\lambda=1$ $\lambda=2$ $\lambda\geqslant 3$ pred. $0.5601$
$0.2801$ $0.1050$ $0.0546$ $18988$ $0.6238$ $0.2538$ $0.0820$ $0.0405$

$p=3$, $ord(\chi)=4$
$N$ $\lambda=0$ $\lambda=1$ $\lambda=2$ $\lambda\geqslant 3$ pred. $0.8766$
$0.1096$ $0.0123$ $0.0015$ $89374$ $0.8880$ $0.1004$ $0.0106$ $0.0011$

$p=3$, $ord(\chi)=8$
$N$ $\lambda=0$ $\lambda=1$ $\lambda=2$ $\lambda\geqslant 3$ pred. $0.8766$
$0.1096$ $0.0123$ $0.0015$ $129600$ $0.8819$ $0.1052$ $0.0114$ $0.0015$

$p=5$, $ord(\chi)=2$
$N$ $\lambda=0$ $\lambda=1$ $\lambda=2$ $\lambda\geqslant 3$ pred. $0.7603$
$0.1901$ $0.0396$ $0.0100$ $17725$ $0.7827$ $0.1737$ $0.0345$ $0.0091$

$p=5$, $ord(\chi)=4$
$N$ $\lambda=0$ $\lambda=1$ $\lambda=2$ $\lambda\geqslant 3$ pred. $0.7603$
$0.1901$ $0.0396$ $0.0100$ $93424$ $0.7760$ $0.1790$ $0.0360$ $0.0090$

$p=7$, $ord(\chi)=4$
$N$ $\lambda=0$ $\lambda=1$ $\lambda=2$ $\lambda\geqslant 3$ pred. $0.9792$
$0.0204$ $0.0004$ $0.0000$ $85862$ $0.9805$ $0.0191$ $0.0004$ $0.0000$

$p=7$, $ord(\chi)=6$
$N$ $\lambda=0$ $\lambda=1$ $\lambda=2$ $\lambda\geqslant 3$ pred. $0.8368$
$0.1395$ $0.0203$ $0.0034$ $266777$ $0.8430$ $0.1343$ $0.0197$ $0.0031$

$p=11$, $ord(\chi)=6$
$N$ $\lambda=0$ $\lambda=1$ $\lambda=2$ $\lambda\geqslant 3$ pred. $0.9917$
$0.0083$ $0.00007$ $0.00000$ $252356$ $0.9924$ $0.00756$ $0.00004$ $0.00000$

Acknowledgments. The author warmly thanks Daniel Delbourgo for helpful
discussions, his support and encouragement.

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