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Properties

Label 1-131-131.20-r0-0-0
Degree $1$
Conductor $131$
Sign $0.584 - 0.811i$
Analytic cond. $0.608361$
Root an. cond. $0.608361$
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  + (−0.681 + 0.732i)2-s + (−0.861 − 0.506i)3-s + (−0.0724 − 0.997i)4-s + (0.995 − 0.0965i)5-s + (0.958 − 0.285i)6-s + (−0.943 + 0.331i)7-s + (0.779 + 0.626i)8-s + (0.485 + 0.873i)9-s + (−0.607 + 0.794i)10-s + (−0.443 − 0.896i)11-s + (−0.443 + 0.896i)12-s + (−0.607 − 0.794i)13-s + (0.399 − 0.916i)14-s + (−0.906 − 0.421i)15-s + (−0.989 + 0.144i)16-s + (0.715 − 0.698i)17-s + ⋯
L(s)  = 1  + (−0.681 + 0.732i)2-s + (−0.861 − 0.506i)3-s + (−0.0724 − 0.997i)4-s + (0.995 − 0.0965i)5-s + (0.958 − 0.285i)6-s + (−0.943 + 0.331i)7-s + (0.779 + 0.626i)8-s + (0.485 + 0.873i)9-s + (−0.607 + 0.794i)10-s + (−0.443 − 0.896i)11-s + (−0.443 + 0.896i)12-s + (−0.607 − 0.794i)13-s + (0.399 − 0.916i)14-s + (−0.906 − 0.421i)15-s + (−0.989 + 0.144i)16-s + (0.715 − 0.698i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 131 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.584 - 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 131 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.584 - 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(131\)
Sign: $0.584 - 0.811i$
Analytic conductor: \(0.608361\)
Root analytic conductor: \(0.608361\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{131} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 131,\ (0:\ ),\ 0.584 - 0.811i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4647473384 - 0.2379046144i\)
\(L(\frac12)\) \(\approx\) \(0.4647473384 - 0.2379046144i\)
\(L(1)\) \(\approx\) \(0.5923243781 - 0.04583792132i\)
\(L(1)\) \(\approx\) \(0.5923243781 - 0.04583792132i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad131 \( 1 \)
good2 \( 1 + (-0.681 + 0.732i)T \)
3 \( 1 + (-0.861 - 0.506i)T \)
5 \( 1 + (0.995 - 0.0965i)T \)
7 \( 1 + (-0.943 + 0.331i)T \)
11 \( 1 + (-0.443 - 0.896i)T \)
13 \( 1 + (-0.607 - 0.794i)T \)
17 \( 1 + (0.715 - 0.698i)T \)
19 \( 1 + (0.885 - 0.464i)T \)
23 \( 1 + (-0.998 + 0.0483i)T \)
29 \( 1 + (0.485 - 0.873i)T \)
31 \( 1 + (-0.262 - 0.964i)T \)
37 \( 1 + (0.644 + 0.764i)T \)
41 \( 1 + (-0.989 - 0.144i)T \)
43 \( 1 + (0.215 - 0.976i)T \)
47 \( 1 + (0.120 - 0.992i)T \)
53 \( 1 + (-0.809 - 0.587i)T \)
59 \( 1 + (-0.168 + 0.985i)T \)
61 \( 1 + (0.309 - 0.951i)T \)
67 \( 1 + (0.215 + 0.976i)T \)
71 \( 1 + (-0.354 + 0.935i)T \)
73 \( 1 + (0.309 + 0.951i)T \)
79 \( 1 + (-0.748 + 0.663i)T \)
83 \( 1 + (0.926 + 0.377i)T \)
89 \( 1 + (-0.809 + 0.587i)T \)
97 \( 1 + (0.0241 + 0.999i)T \)
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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

−28.80166084391936857587696374011, −28.20496087191090938105167826490, −26.86010801852171539341827578525, −26.12498031076121444357092600284, −25.28806079925163250789563357923, −23.57485772806566794969854806066, −22.42564089020111040765161190829, −21.775931505454829265525835793017, −20.8705961250210921816804886368, −19.80212816614525343853671466091, −18.45743886471680947214233760902, −17.6928758534368842111519531292, −16.73938404547115627376531022054, −16.05058752714175782279490619168, −14.25385989082099585262535585810, −12.77459995586061656776482551558, −12.15125510752471375702238565642, −10.612014609985492440959399528221, −9.93899981331812395329631230232, −9.34258941715471162069457878037, −7.35432952537490180900331818953, −6.210133099178459630587596964429, −4.69632211964589971423502589856, −3.252438730757434134895301337187, −1.60004602982039797075322738616, 0.68080246206717958110423274567, 2.50722585806510530648074542859, 5.314499102812151603681368317892, 5.805828361483398917340888549586, 6.88734176305079714360920223146, 8.10864623285099137317420584309, 9.669545097406412038278626897497, 10.24172278735823407401734341356, 11.72835178376023919584924343620, 13.15815803347869285101855990390, 13.90261117359283997730381932234, 15.59942107191430939233874193597, 16.461387016020411167317416208372, 17.27376626309884179308454325953, 18.29645587275985192425653469882, 18.87102427428448918032094220050, 20.205433822735411704502090790389, 21.92077073472778249921948201209, 22.56867469641745703227182221593, 23.799611467006817449190385296264, 24.71551652789097809281001477141, 25.36481996649456398801564571146, 26.44866292192254105468696278859, 27.61219400329542144414946381497, 28.69276669295536515469517907634

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