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Properties

Label 1-131-131.15-r0-0-0
Degree $1$
Conductor $131$
Sign $0.605 - 0.795i$
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  + (0.836 − 0.548i)2-s + (−0.607 + 0.794i)3-s + (0.399 − 0.916i)4-s + (0.0241 − 0.999i)5-s + (−0.0724 + 0.997i)6-s + (0.644 + 0.764i)7-s + (−0.168 − 0.985i)8-s + (−0.262 − 0.964i)9-s + (−0.527 − 0.849i)10-s + (0.485 − 0.873i)11-s + (0.485 + 0.873i)12-s + (−0.527 + 0.849i)13-s + (0.958 + 0.285i)14-s + (0.779 + 0.626i)15-s + (−0.681 − 0.732i)16-s + (0.981 + 0.192i)17-s + ⋯
L(s)  = 1  + (0.836 − 0.548i)2-s + (−0.607 + 0.794i)3-s + (0.399 − 0.916i)4-s + (0.0241 − 0.999i)5-s + (−0.0724 + 0.997i)6-s + (0.644 + 0.764i)7-s + (−0.168 − 0.985i)8-s + (−0.262 − 0.964i)9-s + (−0.527 − 0.849i)10-s + (0.485 − 0.873i)11-s + (0.485 + 0.873i)12-s + (−0.527 + 0.849i)13-s + (0.958 + 0.285i)14-s + (0.779 + 0.626i)15-s + (−0.681 − 0.732i)16-s + (0.981 + 0.192i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 131 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 131 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(131\)
Sign: $0.605 - 0.795i$
Analytic conductor: \(0.608361\)
Root analytic conductor: \(0.608361\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{131} (15, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 131,\ (0:\ ),\ 0.605 - 0.795i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.319916593 - 0.6540854480i\)
\(L(\frac12)\) \(\approx\) \(1.319916593 - 0.6540854480i\)
\(L(1)\) \(\approx\) \(1.345742447 - 0.4128615412i\)
\(L(1)\) \(\approx\) \(1.345742447 - 0.4128615412i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad131 \( 1 \)
good2 \( 1 + (0.836 - 0.548i)T \)
3 \( 1 + (-0.607 + 0.794i)T \)
5 \( 1 + (0.0241 - 0.999i)T \)
7 \( 1 + (0.644 + 0.764i)T \)
11 \( 1 + (0.485 - 0.873i)T \)
13 \( 1 + (-0.527 + 0.849i)T \)
17 \( 1 + (0.981 + 0.192i)T \)
19 \( 1 + (0.120 - 0.992i)T \)
23 \( 1 + (0.715 - 0.698i)T \)
29 \( 1 + (-0.262 + 0.964i)T \)
31 \( 1 + (-0.443 + 0.896i)T \)
37 \( 1 + (0.215 + 0.976i)T \)
41 \( 1 + (-0.681 + 0.732i)T \)
43 \( 1 + (-0.943 - 0.331i)T \)
47 \( 1 + (-0.354 + 0.935i)T \)
53 \( 1 + (-0.809 - 0.587i)T \)
59 \( 1 + (-0.906 + 0.421i)T \)
61 \( 1 + (0.309 - 0.951i)T \)
67 \( 1 + (-0.943 + 0.331i)T \)
71 \( 1 + (0.885 - 0.464i)T \)
73 \( 1 + (0.309 + 0.951i)T \)
79 \( 1 + (0.568 + 0.822i)T \)
83 \( 1 + (0.995 - 0.0965i)T \)
89 \( 1 + (-0.809 + 0.587i)T \)
97 \( 1 + (0.926 - 0.377i)T \)
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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

−29.47171497592408721113866266242, −27.70703832108368459574901397719, −26.75012993091836312477954714187, −25.40383474491931048076930803414, −24.85917208038865687090443456819, −23.54633613563394433648151799630, −22.9636344810364095911732887956, −22.31806878555065749327055734569, −20.98924615642144959906437820556, −19.787179794467501010103535832962, −18.38553873566017283413910569567, −17.42232704269629466791497737732, −16.80000254088703800949619539652, −15.12082830978250676540369131844, −14.37781474544910695588985034195, −13.39947750777698918238964602577, −12.21560701076466038852480994911, −11.360309607577294562461270636, −10.18581332418696788358125853508, −7.60760730022433175549309998766, −7.507814515242564702404256497, −6.17154927023883488561136145994, −5.07590535321213104841914608118, −3.56755985298844270014116134089, −1.968143440686342373711746343192, 1.348331069578387675933790310261, 3.250379698063899047059117634738, 4.72464836920023906092399564298, 5.20272413129933053668640585843, 6.45419566459071124776683865518, 8.72602354719784362314999835914, 9.59170845752083705706175499537, 11.05921016883230915143120562307, 11.80532953521236437093244468686, 12.62584033259043409689500355287, 14.14035346093488776123917478852, 15.043770483312534697495441591194, 16.20343282947095257855448699962, 16.977816077126634310809843830999, 18.52207091824731369276319395010, 19.765932446354107545123580455961, 20.87617370790500276254954958290, 21.56398843912708783960391440325, 22.12594246156149667576979180471, 23.65204864947350080820500367454, 24.123525768631986667678560769871, 25.2355505265185278233539256332, 27.0425358898654509445811683095, 27.799934252414909243822405184923, 28.618177485196660287724694339716

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