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Properties

Label 1-5e2-25.14-r0-0-0
Degree $1$
Conductor $25$
Sign $0.187 + 0.982i$
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  + (−0.309 + 0.951i)2-s + (0.809 + 0.587i)3-s + (−0.809 − 0.587i)4-s + (−0.809 + 0.587i)6-s − 7-s + (0.809 − 0.587i)8-s + (0.309 + 0.951i)9-s + (0.309 − 0.951i)11-s + (−0.309 − 0.951i)12-s + (−0.309 − 0.951i)13-s + (0.309 − 0.951i)14-s + (0.309 + 0.951i)16-s + (0.809 − 0.587i)17-s − 18-s + (−0.809 + 0.587i)19-s + ⋯
L(s)  = 1  + (−0.309 + 0.951i)2-s + (0.809 + 0.587i)3-s + (−0.809 − 0.587i)4-s + (−0.809 + 0.587i)6-s − 7-s + (0.809 − 0.587i)8-s + (0.309 + 0.951i)9-s + (0.309 − 0.951i)11-s + (−0.309 − 0.951i)12-s + (−0.309 − 0.951i)13-s + (0.309 − 0.951i)14-s + (0.309 + 0.951i)16-s + (0.809 − 0.587i)17-s − 18-s + (−0.809 + 0.587i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.187 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.187 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(25\)    =    \(5^{2}\)
Sign: $0.187 + 0.982i$
Analytic conductor: \(0.116099\)
Root analytic conductor: \(0.116099\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{25} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 25,\ (0:\ ),\ 0.187 + 0.982i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5240838236 + 0.4335598446i\)
\(L(\frac12)\) \(\approx\) \(0.5240838236 + 0.4335598446i\)
\(L(1)\) \(\approx\) \(0.7706990758 + 0.4515038146i\)
\(L(1)\) \(\approx\) \(0.7706990758 + 0.4515038146i\)

Euler product

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

−38.28063243031528312009180392797, −36.692984174110792567464581601384, −36.05393354849720361244723971653, −34.887120850584521875781093352581, −32.53423775786614196310832036417, −31.45895811645762387438683159672, −30.345757362397796713437597878831, −29.29868373787037724456850028721, −28.03366939887295734630480981476, −26.28598245387479879441431448339, −25.58333562770774913914970287131, −23.61413591520153963594981421375, −22.0873815233214598442836214081, −20.560065527476893369493659005480, −19.48026988227819791926043054632, −18.59221496591904323290261617352, −16.92435450015065411443429305569, −14.64851488374259577660623929047, −13.108533409221915032416646344925, −12.15610361504569240431112275340, −9.984614757569034276556628636003, −8.83004689737836649629298973739, −7.0252574621156174444793516646, −3.87162723584480278913458272457, −2.13858987186166861971624060558, 3.56286301215718626248462418835, 5.69525797854698655509075110454, 7.65721871714893782562673106625, 9.1007059557284357333270928515, 10.2649821979982712510127350133, 13.17474937252591201691648066386, 14.47471425031826237727039410793, 15.7750222662889525951832674177, 16.77026179013987180530517255610, 18.779510872615321987949064088151, 19.815615669641949688948559710507, 21.74419316688276455445293419541, 23.02881841425419766842900148042, 24.83111727710086347653701125163, 25.647646736090431891325639246723, 26.851369901539873776928623922210, 27.81613798359506846716938392087, 29.72059632287232218730806481137, 31.863185453142629020134564875139, 32.142785570811738917279883934617, 33.51027636896173048414152156095, 34.91139412540518781839425671576, 36.11669324804437190080436291140, 37.26380364511621541164631254905, 38.36385625028819045647228142415

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