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Properties

Label 1-5e2-25.21-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: $\chi_{25} (21, \cdot )$
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

−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