Properties

Degree 1
Conductor 131
Sign $0.605 - 0.795i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(131\)
\( \varepsilon \)  =  $0.605 - 0.795i$
motivic weight  =  \(0\)
character  :  $\chi_{131} (15, \cdot )$
Sato-Tate  :  $\mu(65)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 131,\ (0:\ ),\ 0.605 - 0.795i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.319916593 - 0.6540854480i$
$L(\frac12,\chi)$  $\approx$  $1.319916593 - 0.6540854480i$
$L(\chi,1)$  $\approx$  1.345742447 - 0.4128615412i
$L(1,\chi)$  $\approx$  1.345742447 - 0.4128615412i

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

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