↑

Properties

Label 1-107-107.8-r1-0-0
Degree $1$
Conductor $107$
Sign $0.531 + 0.847i$
Analytic cond. $11.4987$
Root an. cond. $11.4987$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.984 + 0.176i)2-s + (0.992 − 0.118i)3-s + (0.937 + 0.348i)4-s + (−0.482 + 0.875i)5-s + (0.998 + 0.0592i)6-s + (0.205 + 0.978i)7-s + (0.861 + 0.508i)8-s + (0.972 − 0.234i)9-s + (−0.630 + 0.776i)10-s + (−0.717 − 0.696i)11-s + (0.972 + 0.234i)12-s + (−0.794 + 0.606i)13-s + (0.0296 + 0.999i)14-s + (−0.375 + 0.926i)15-s + (0.757 + 0.652i)16-s + (0.430 − 0.902i)17-s + ⋯
L(s)  = 1  + (0.984 + 0.176i)2-s + (0.992 − 0.118i)3-s + (0.937 + 0.348i)4-s + (−0.482 + 0.875i)5-s + (0.998 + 0.0592i)6-s + (0.205 + 0.978i)7-s + (0.861 + 0.508i)8-s + (0.972 − 0.234i)9-s + (−0.630 + 0.776i)10-s + (−0.717 − 0.696i)11-s + (0.972 + 0.234i)12-s + (−0.794 + 0.606i)13-s + (0.0296 + 0.999i)14-s + (−0.375 + 0.926i)15-s + (0.757 + 0.652i)16-s + (0.430 − 0.902i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 107 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.531 + 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 107 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.531 + 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(107\)
Sign: $0.531 + 0.847i$
Analytic conductor: \(11.4987\)
Root analytic conductor: \(11.4987\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{107} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 107,\ (1:\ ),\ 0.531 + 0.847i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.595234266 + 1.988864295i\)
\(L(\frac12)\) \(\approx\) \(3.595234266 + 1.988864295i\)
\(L(1)\) \(\approx\) \(2.351187043 + 0.7505014052i\)
\(L(1)\) \(\approx\) \(2.351187043 + 0.7505014052i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad107 \( 1 \)
good2 \( 1 + (0.984 + 0.176i)T \)
3 \( 1 + (0.992 - 0.118i)T \)
5 \( 1 + (-0.482 + 0.875i)T \)
7 \( 1 + (0.205 + 0.978i)T \)
11 \( 1 + (-0.717 - 0.696i)T \)
13 \( 1 + (-0.794 + 0.606i)T \)
17 \( 1 + (0.430 - 0.902i)T \)
19 \( 1 + (0.263 + 0.964i)T \)
23 \( 1 + (0.0296 - 0.999i)T \)
29 \( 1 + (0.829 - 0.558i)T \)
31 \( 1 + (0.0887 - 0.996i)T \)
37 \( 1 + (0.889 + 0.456i)T \)
41 \( 1 + (0.674 + 0.737i)T \)
43 \( 1 + (-0.482 - 0.875i)T \)
47 \( 1 + (0.674 - 0.737i)T \)
53 \( 1 + (-0.984 + 0.176i)T \)
59 \( 1 + (-0.829 - 0.558i)T \)
61 \( 1 + (-0.205 + 0.978i)T \)
67 \( 1 + (0.861 - 0.508i)T \)
71 \( 1 + (-0.992 - 0.118i)T \)
73 \( 1 + (-0.582 + 0.812i)T \)
79 \( 1 + (0.147 - 0.989i)T \)
83 \( 1 + (-0.915 - 0.403i)T \)
89 \( 1 + (-0.998 + 0.0592i)T \)
97 \( 1 + (0.630 - 0.776i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−29.52325777474312275391531031801, −28.27012754743771194498552272546, −27.14601625640194171712332284511, −25.91884686214260016821465211237, −24.92959454259407987310258043960, −23.889631050214717270544287227367, −23.315496676555056132129334001673, −21.70110389011848693332292056918, −20.81639375007721140112896470242, −19.90411676303066771405496418988, −19.56213053099717459792974007582, −17.464524071209366494273619799510, −16.05656426855231797435065104835, −15.257167658114903208318026847925, −14.19707862071987304694582715213, −13.08625884038874680064528672403, −12.45800817301914817689433698223, −10.78675762072142904250050261605, −9.69613830468268079503556878728, −7.94641637883783491170200755636, −7.24972696822937463487001446911, −5.07664208876633793023486732689, −4.2522351269646066882925654838, −2.98730305537473368981679835702, −1.346506934257060359298918731041, 2.376329745337208249831311158754, 3.04940083666485756141648911306, 4.51434901505020347993733594011, 6.09428133013802639806152060091, 7.41839261341344939300246424022, 8.27319722540432298428748208197, 10.00958079250708046621968146744, 11.51626142647995908028068130029, 12.41913196986914594521760565984, 13.83157471048098949493588003695, 14.56407249283964849605109615516, 15.380432056576708861772582539789, 16.338168619496161145761762011013, 18.47268018216549593341251251760, 19.04789156174668683853261004448, 20.39257526383610515338495008733, 21.37576108464297367913820244222, 22.17648676365608578807316807844, 23.40604748268684374669547753743, 24.48843360208372411640281067895, 25.185655675835598780854509495386, 26.34696479772687202735130173095, 27.07298367533878085562312620901, 28.9125922058527461210461248239, 29.87785986887906419169112061369

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