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Properties

Label 1-5e2-25.6-r0-0-0
Degree $1$
Conductor $25$
Sign $0.968 - 0.248i$
Analytic cond. $0.116099$
Root an. cond. $0.116099$
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  + (−0.809 + 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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(25\)    =    \(5^{2}\)
Sign: $0.968 - 0.248i$
Analytic conductor: \(0.116099\)
Root analytic conductor: \(0.116099\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{25} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 25,\ (0:\ ),\ 0.968 - 0.248i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5537986884 - 0.06996104409i\)
\(L(\frac12)\) \(\approx\) \(0.5537986884 - 0.06996104409i\)
\(L(1)\) \(\approx\) \(0.7399223180 - 0.04332015061i\)
\(L(1)\) \(\approx\) \(0.7399223180 - 0.04332015061i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
good2 \( 1 + (-0.809 + 0.587i)T \)
3 \( 1 + (0.309 - 0.951i)T \)
7 \( 1 + T \)
11 \( 1 + (-0.809 + 0.587i)T \)
13 \( 1 + (-0.809 - 0.587i)T \)
17 \( 1 + (0.309 + 0.951i)T \)
19 \( 1 + (0.309 + 0.951i)T \)
23 \( 1 + (-0.809 + 0.587i)T \)
29 \( 1 + (0.309 - 0.951i)T \)
31 \( 1 + (0.309 + 0.951i)T \)
37 \( 1 + (-0.809 - 0.587i)T \)
41 \( 1 + (-0.809 - 0.587i)T \)
43 \( 1 + T \)
47 \( 1 + (0.309 - 0.951i)T \)
53 \( 1 + (0.309 - 0.951i)T \)
59 \( 1 + (-0.809 - 0.587i)T \)
61 \( 1 + (-0.809 + 0.587i)T \)
67 \( 1 + (0.309 + 0.951i)T \)
71 \( 1 + (0.309 - 0.951i)T \)
73 \( 1 + (-0.809 + 0.587i)T \)
79 \( 1 + (0.309 - 0.951i)T \)
83 \( 1 + (0.309 + 0.951i)T \)
89 \( 1 + (-0.809 + 0.587i)T \)
97 \( 1 + (0.309 - 0.951i)T \)
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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

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

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