Properties

Degree 1
Conductor $ 5^{2} $
Sign $0.968 + 0.248i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.809 − 0.587i)2-s + (0.309 + 0.951i)3-s + (0.309 + 0.951i)4-s + (0.309 − 0.951i)6-s + 7-s + (0.309 − 0.951i)8-s + (−0.809 + 0.587i)9-s + (−0.809 − 0.587i)11-s + (−0.809 + 0.587i)12-s + (−0.809 + 0.587i)13-s + (−0.809 − 0.587i)14-s + (−0.809 + 0.587i)16-s + (0.309 − 0.951i)17-s + 18-s + (0.309 − 0.951i)19-s + ⋯
L(s,χ)  = 1  + (−0.809 − 0.587i)2-s + (0.309 + 0.951i)3-s + (0.309 + 0.951i)4-s + (0.309 − 0.951i)6-s + 7-s + (0.309 − 0.951i)8-s + (−0.809 + 0.587i)9-s + (−0.809 − 0.587i)11-s + (−0.809 + 0.587i)12-s + (−0.809 + 0.587i)13-s + (−0.809 − 0.587i)14-s + (−0.809 + 0.587i)16-s + (0.309 − 0.951i)17-s + 18-s + (0.309 − 0.951i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.968 + 0.248i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 25 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.968 + 0.248i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(25\)    =    \(5^{2}\)
\( \varepsilon \)  =  $0.968 + 0.248i$
motivic weight  =  \(0\)
character  :  $\chi_{25} (21, \cdot )$
Sato-Tate  :  $\mu(5)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 25,\ (0:\ ),\ 0.968 + 0.248i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.5537986884 + 0.06996104409i$
$L(\frac12,\chi)$  $\approx$  $0.5537986884 + 0.06996104409i$
$L(\chi,1)$  $\approx$  0.7399223180 + 0.04332015061i
$L(1,\chi)$  $\approx$  0.7399223180 + 0.04332015061i

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

−37.71873028682921110349889624356, −36.84276021213698190862808901588, −35.9522251515344351871917654801, −34.61478930971423964432276944461, −33.77008020972040341009485060611, −32.007189998043253669036990006521, −30.5807006893630825131987600964, −29.25123167701774503367530465595, −27.942433214485561456210719787170, −26.53257581262411383545971291802, −25.269239075853214896309092337709, −24.269040034525302538322870017392, −23.236008705864727240458110542953, −20.70534485201465709164227493774, −19.43452946084196907421018968771, −18.0975185714585925102348961302, −17.30902994093537367573409417979, −15.25020791182264383439566455997, −14.089641796943627977988788343479, −12.140976563476839697183614421486, −10.26126692143616249840620923008, −8.27993401244947581823398314674, −7.45071687788618412374063094684, −5.55340258128875097150319382663, −1.91294097474225892842044179076, 2.68355491112275912524542532413, 4.72842543600737728591551864982, 7.78096183160697780828333217129, 9.12534302713112165414639265240, 10.5393274086782915735324062917, 11.71765888521375677458442558269, 13.97503098677160372066480513312, 15.72827701583520332387464016441, 17.0101450651913488266142000195, 18.48726199565144191890169759070, 20.07524325040731827258118812377, 21.084300192480879376603830692506, 22.06893732997405698690320809197, 24.35816262198533611083160840807, 26.0336516921797324530554457269, 26.935333847517968109908365089372, 27.913625349643426474355648429118, 29.2144525004005410230114758316, 30.79819291497720633192514944156, 31.846610265774308262259327195116, 33.73741985988972408028601000199, 34.48945912271288503775684179619, 36.52654463793862044969678031444, 37.09444485231264601170753861285, 38.378086136192828804856037198373

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