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Properties

Label 1-107-107.10-r0-0-0
Degree $1$
Conductor $107$
Sign $0.188 + 0.981i$
Analytic cond. $0.496905$
Root an. cond. $0.496905$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.956 + 0.292i)2-s + (−0.320 − 0.947i)3-s + (0.829 − 0.558i)4-s + (−0.205 + 0.978i)5-s + (0.582 + 0.812i)6-s + (−0.984 + 0.176i)7-s + (−0.630 + 0.776i)8-s + (−0.794 + 0.606i)9-s + (−0.0887 − 0.996i)10-s + (0.972 − 0.234i)11-s + (−0.794 − 0.606i)12-s + (−0.533 + 0.845i)13-s + (0.889 − 0.456i)14-s + (0.992 − 0.118i)15-s + (0.375 − 0.926i)16-s + (0.674 + 0.737i)17-s + ⋯
L(s)  = 1  + (−0.956 + 0.292i)2-s + (−0.320 − 0.947i)3-s + (0.829 − 0.558i)4-s + (−0.205 + 0.978i)5-s + (0.582 + 0.812i)6-s + (−0.984 + 0.176i)7-s + (−0.630 + 0.776i)8-s + (−0.794 + 0.606i)9-s + (−0.0887 − 0.996i)10-s + (0.972 − 0.234i)11-s + (−0.794 − 0.606i)12-s + (−0.533 + 0.845i)13-s + (0.889 − 0.456i)14-s + (0.992 − 0.118i)15-s + (0.375 − 0.926i)16-s + (0.674 + 0.737i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 107 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.188 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 107 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.188 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(107\)
Sign: $0.188 + 0.981i$
Analytic conductor: \(0.496905\)
Root analytic conductor: \(0.496905\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{107} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 107,\ (0:\ ),\ 0.188 + 0.981i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3296864527 + 0.2723121582i\)
\(L(\frac12)\) \(\approx\) \(0.3296864527 + 0.2723121582i\)
\(L(1)\) \(\approx\) \(0.5181693188 + 0.1089287204i\)
\(L(1)\) \(\approx\) \(0.5181693188 + 0.1089287204i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad107 \( 1 \)
good2 \( 1 + (-0.956 + 0.292i)T \)
3 \( 1 + (-0.320 - 0.947i)T \)
5 \( 1 + (-0.205 + 0.978i)T \)
7 \( 1 + (-0.984 + 0.176i)T \)
11 \( 1 + (0.972 - 0.234i)T \)
13 \( 1 + (-0.533 + 0.845i)T \)
17 \( 1 + (0.674 + 0.737i)T \)
19 \( 1 + (-0.430 + 0.902i)T \)
23 \( 1 + (0.889 + 0.456i)T \)
29 \( 1 + (-0.998 + 0.0592i)T \)
31 \( 1 + (0.147 + 0.989i)T \)
37 \( 1 + (0.263 + 0.964i)T \)
41 \( 1 + (0.757 + 0.652i)T \)
43 \( 1 + (-0.205 - 0.978i)T \)
47 \( 1 + (0.757 - 0.652i)T \)
53 \( 1 + (-0.956 - 0.292i)T \)
59 \( 1 + (-0.998 - 0.0592i)T \)
61 \( 1 + (-0.984 - 0.176i)T \)
67 \( 1 + (-0.630 - 0.776i)T \)
71 \( 1 + (-0.320 + 0.947i)T \)
73 \( 1 + (-0.861 - 0.508i)T \)
79 \( 1 + (-0.717 + 0.696i)T \)
83 \( 1 + (0.937 + 0.348i)T \)
89 \( 1 + (0.582 - 0.812i)T \)
97 \( 1 + (-0.0887 - 0.996i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−29.11648889911098499767205944869, −28.076862869198970744607374307603, −27.60898633471161285490743630659, −26.58325758909012951941223966764, −25.50628147089309227078316255496, −24.61147894498585428968382091092, −22.978510311328970816023432750311, −22.02297604092890439121327918256, −20.76565693842444512202194964529, −20.07400601223231062525531007349, −19.20198079879101944422466855031, −17.45165673784679192567901706680, −16.81726333560874420250389252738, −16.02081560465087099465640947177, −14.99372148622114611335833285495, −12.891187824264353701467977028406, −11.96635433125981579946247316064, −10.77533115704577465149049907979, −9.44458400156078519238846619042, −9.166892377774737278317863194953, −7.48115701119360531391329018339, −5.952475915658355610448919125082, −4.334896396371039064222145232639, −2.9985901985055055330365496956, −0.60119777593264802300864322754, 1.69231334611731000233929111523, 3.20648077472858124446159596736, 5.97818425786329617101904909795, 6.668936425125890447946044685604, 7.57247101332870845797907083719, 8.99830800850530993496165225779, 10.26504655233426281343830723116, 11.45066650551134448065813194822, 12.3686270912883590305183541886, 14.08071453449601231370985476942, 15.01866897616281304625562533074, 16.57200004696871029705733666400, 17.19153900443675834424306796074, 18.66690301230322929618274454350, 19.03809597030150127321748060191, 19.78369080763404760317011469276, 21.72275968236529587451587085004, 22.903378364518516148611817853779, 23.75342510498769770313968744289, 25.03399926829814327025780615356, 25.65390423178891009722894021807, 26.70417823671814738206332543213, 27.78042126518212104318331524912, 28.9781147910573325997003288130, 29.59027413588867728433843836181

Graph of the $Z$-function along the critical line