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Properties

Label 1-131-131.25-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  + (−0.262 − 0.964i)2-s + (0.958 − 0.285i)3-s + (−0.861 + 0.506i)4-s + (−0.943 − 0.331i)5-s + (−0.527 − 0.849i)6-s + (0.926 − 0.377i)7-s + (0.715 + 0.698i)8-s + (0.836 − 0.548i)9-s + (−0.0724 + 0.997i)10-s + (−0.681 − 0.732i)11-s + (−0.681 + 0.732i)12-s + (−0.0724 − 0.997i)13-s + (−0.607 − 0.794i)14-s + (−0.998 − 0.0483i)15-s + (0.485 − 0.873i)16-s + (−0.906 + 0.421i)17-s + ⋯
L(s)  = 1  + (−0.262 − 0.964i)2-s + (0.958 − 0.285i)3-s + (−0.861 + 0.506i)4-s + (−0.943 − 0.331i)5-s + (−0.527 − 0.849i)6-s + (0.926 − 0.377i)7-s + (0.715 + 0.698i)8-s + (0.836 − 0.548i)9-s + (−0.0724 + 0.997i)10-s + (−0.681 − 0.732i)11-s + (−0.681 + 0.732i)12-s + (−0.0724 − 0.997i)13-s + (−0.607 − 0.794i)14-s + (−0.998 − 0.0483i)15-s + (0.485 − 0.873i)16-s + (−0.906 + 0.421i)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: $\chi_{131} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 131,\ (0:\ ),\ -0.584 - 0.811i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4746884886 - 0.9273660449i\)
\(L(\frac12)\) \(\approx\) \(0.4746884886 - 0.9273660449i\)
\(L(1)\) \(\approx\) \(0.7863425241 - 0.6765780780i\)
\(L(1)\) \(\approx\) \(0.7863425241 - 0.6765780780i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad131 \( 1 \)
good2 \( 1 + (-0.262 - 0.964i)T \)
3 \( 1 + (0.958 - 0.285i)T \)
5 \( 1 + (-0.943 - 0.331i)T \)
7 \( 1 + (0.926 - 0.377i)T \)
11 \( 1 + (-0.681 - 0.732i)T \)
13 \( 1 + (-0.0724 - 0.997i)T \)
17 \( 1 + (-0.906 + 0.421i)T \)
19 \( 1 + (0.120 - 0.992i)T \)
23 \( 1 + (-0.168 + 0.985i)T \)
29 \( 1 + (0.836 + 0.548i)T \)
31 \( 1 + (-0.989 - 0.144i)T \)
37 \( 1 + (0.995 + 0.0965i)T \)
41 \( 1 + (0.485 + 0.873i)T \)
43 \( 1 + (0.0241 - 0.999i)T \)
47 \( 1 + (-0.354 + 0.935i)T \)
53 \( 1 + (-0.809 + 0.587i)T \)
59 \( 1 + (0.981 + 0.192i)T \)
61 \( 1 + (0.309 + 0.951i)T \)
67 \( 1 + (0.0241 + 0.999i)T \)
71 \( 1 + (0.885 - 0.464i)T \)
73 \( 1 + (0.309 - 0.951i)T \)
79 \( 1 + (0.568 + 0.822i)T \)
83 \( 1 + (0.215 - 0.976i)T \)
89 \( 1 + (-0.809 - 0.587i)T \)
97 \( 1 + (0.644 + 0.764i)T \)
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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

−28.64669372498539591869002510654, −27.57347618090339794211429671280, −26.82948226557908365472393936193, −26.222310021697856182996491647444, −25.049471761608317601005336327444, −24.24734651686956595342747998882, −23.331052546156015822492998214378, −22.14216518597764979043872548738, −20.88603016109956909028210348426, −19.79961878013041993070037900460, −18.68971684525597881122709570366, −18.07075183453534676917264995654, −16.39332030289510231510437278064, −15.57967186983493135833826806737, −14.72234411712236910240211305387, −14.10500148951097149438709742874, −12.59269670719336032648834177526, −11.01014009558118450379505197303, −9.71565978127944685111439568475, −8.50847268828578647974138067041, −7.84065527028669614107425329238, −6.82626258451210226782205705649, −4.861730146508515533333800191571, −4.093334390547271788010381213294, −2.15849980317827096311636747476, 1.036303426644573767826912359568, 2.653439871046725931769358402208, 3.78523114260525264586734166549, 4.92414442582714049248640922684, 7.53111884156312444692361116903, 8.16656063145271757704130360848, 9.048025891914000675272215615817, 10.6190668716814164008258235721, 11.458047802067830721695260843370, 12.81347008124064462590936579509, 13.49728465457928390984615961640, 14.785790795644787712465421701726, 15.89317544222510192942551772829, 17.551209627213141655756603528288, 18.36537182705084733409369253551, 19.62792806381835707033110844264, 20.019148218422784120139900296557, 20.96725692595274750977620250075, 21.955478939670475310322977681615, 23.5964835524188471786457271875, 24.09178361764799395153996670854, 25.57355473834443508482988771600, 26.84004328291787406569875227996, 27.1142401711802252690233384124, 28.26890392795039194088579983591

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