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Properties

Label 1-107-107.40-r0-0-0
Degree $1$
Conductor $107$
Sign $0.411 + 0.911i$
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  + (−0.984 + 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 107 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.411 + 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 107 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.411 + 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(107\)
Sign: $0.411 + 0.911i$
Analytic conductor: \(0.496905\)
Root analytic conductor: \(0.496905\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{107} (40, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 107,\ (0:\ ),\ 0.411 + 0.911i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7869712759 + 0.5083602975i\)
\(L(\frac12)\) \(\approx\) \(0.7869712759 + 0.5083602975i\)
\(L(1)\) \(\approx\) \(0.8926332400 + 0.3219876593i\)
\(L(1)\) \(\approx\) \(0.8926332400 + 0.3219876593i\)

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 \( 1 + (-0.630 - 0.776i)T \)
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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

−29.2431543412113042318710849292, −28.61008475008693352909614918595, −26.93334930579115106447290305116, −26.610828797997077742421309544, −25.49180433122557167438813183950, −24.53989053259638226673439046523, −23.80917542910804979639941623547, −21.562457681615838849843818118282, −20.85742377865796681922785060614, −19.931709034926120625208901924332, −19.210577708121094467895924218812, −17.96498343652946230126594460207, −16.75520093551342337577244641720, −16.06730329860152886766412374003, −14.49465696465007544218899152545, −13.30368469195968806456409191648, −12.3442719770582885024813163331, −10.52042413146584209654647701214, −9.71733513034192292794229838767, −8.565483825877370020060682101365, −7.76997091035830435733494813840, −6.389127273429455639575391758862, −4.2513241007281983180830774521, −2.65741759083812144419970059221, −1.25941774091594170188471622713, 2.31158077653381620349155218408, 2.799944061985931477948874359959, 5.34264966822128614451348875136, 6.95393663419253681155707591012, 7.78469341527184192369357531606, 9.28307452271907869534363273135, 9.78471475246831318444481184375, 11.07439693912450995062461526302, 12.65465070439804056325119112927, 14.13601930430704827109324834752, 15.332349030198693044962277958923, 15.64066717348544955585772543213, 17.61960033123078077175648321183, 18.2927591109659769417447781616, 19.24772862193243084613471270050, 20.201751086994875675072933006, 21.32600524225468652419762049463, 22.33548176845863177010167311285, 24.114179177738178054423844721959, 25.28353459946251131029661510857, 25.603855274344872455462658340304, 26.6365034893579418007725664886, 27.48455487253042982799918563573, 28.7480216161660488651121125024, 29.68697569175949518035701680366

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