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Properties

Label 1-925-925.471-r0-0-0
Degree $1$
Conductor $925$
Sign $-0.783 + 0.621i$
Analytic cond. $4.29568$
Root an. cond. $4.29568$
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  + (0.104 − 0.994i)2-s + (−0.978 − 0.207i)3-s + (−0.978 − 0.207i)4-s + (−0.309 + 0.951i)6-s + (−0.5 + 0.866i)7-s + (−0.309 + 0.951i)8-s + (0.913 + 0.406i)9-s + (−0.809 − 0.587i)11-s + (0.913 + 0.406i)12-s + (0.104 + 0.994i)13-s + (0.809 + 0.587i)14-s + (0.913 + 0.406i)16-s + (0.978 − 0.207i)17-s + (0.5 − 0.866i)18-s + (−0.669 − 0.743i)19-s + ⋯
L(s)  = 1  + (0.104 − 0.994i)2-s + (−0.978 − 0.207i)3-s + (−0.978 − 0.207i)4-s + (−0.309 + 0.951i)6-s + (−0.5 + 0.866i)7-s + (−0.309 + 0.951i)8-s + (0.913 + 0.406i)9-s + (−0.809 − 0.587i)11-s + (0.913 + 0.406i)12-s + (0.104 + 0.994i)13-s + (0.809 + 0.587i)14-s + (0.913 + 0.406i)16-s + (0.978 − 0.207i)17-s + (0.5 − 0.866i)18-s + (−0.669 − 0.743i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.783 + 0.621i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.783 + 0.621i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(925\)    =    \(5^{2} \cdot 37\)
Sign: $-0.783 + 0.621i$
Analytic conductor: \(4.29568\)
Root analytic conductor: \(4.29568\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{925} (471, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 925,\ (0:\ ),\ -0.783 + 0.621i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.04605718700 - 0.1320479461i\)
\(L(\frac12)\) \(\approx\) \(-0.04605718700 - 0.1320479461i\)
\(L(1)\) \(\approx\) \(0.5087425490 - 0.2611722860i\)
\(L(1)\) \(\approx\) \(0.5087425490 - 0.2611722860i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
37 \( 1 \)
good2 \( 1 + (0.104 - 0.994i)T \)
3 \( 1 + (-0.978 - 0.207i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (-0.809 - 0.587i)T \)
13 \( 1 + (0.104 + 0.994i)T \)
17 \( 1 + (0.978 - 0.207i)T \)
19 \( 1 + (-0.669 - 0.743i)T \)
23 \( 1 + (0.809 + 0.587i)T \)
29 \( 1 + (-0.309 - 0.951i)T \)
31 \( 1 + (-0.309 + 0.951i)T \)
41 \( 1 + (-0.104 - 0.994i)T \)
43 \( 1 - T \)
47 \( 1 + (0.309 + 0.951i)T \)
53 \( 1 + (0.669 - 0.743i)T \)
59 \( 1 + (-0.913 - 0.406i)T \)
61 \( 1 + (-0.913 + 0.406i)T \)
67 \( 1 + (0.669 + 0.743i)T \)
71 \( 1 + (-0.978 - 0.207i)T \)
73 \( 1 + (-0.809 - 0.587i)T \)
79 \( 1 + (0.978 + 0.207i)T \)
83 \( 1 + (-0.978 + 0.207i)T \)
89 \( 1 + (0.104 - 0.994i)T \)
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

−22.751166680219673928580243962100, −21.80126789411405201497672131378, −20.96087756576295063413901549929, −20.08293966608480203478683168192, −18.65935019875810890729840711362, −18.34982850261382512633849062097, −17.24708488737562611400526201138, −16.794095640129046403098595243417, −16.20576738411664860307808209089, −15.210617062692343538779511867, −14.73109954587719838597193670318, −13.36266521845039967178972467818, −12.86170535630008533518362946430, −12.19498192382812288333318799785, −10.63316227929832585450409724394, −10.30068525822391492839966750826, −9.42780450768291147293694652789, −8.07563627112699259087010234908, −7.41220030066760266849187654844, −6.58853026398340067982847729636, −5.73279774358954527445890962231, −5.02351814886084926390154845343, −4.11325974346028008521734991387, −3.20644866189878336583750161305, −1.157692961463496038899821292318, 0.07938657440384017577460690740, 1.448559196159897574271092675025, 2.48479908557017356102379759398, 3.4597548384027928973825083657, 4.669917699200748352967185356651, 5.42475104717803134648259573457, 6.12820943947327576385628006485, 7.2775165472718910374853704379, 8.52210145002175509540652767038, 9.337838286565134394288378559733, 10.19518221426232740990202897830, 11.047990794860395609037959642612, 11.67712164916828328368594489532, 12.392370271563076634181885687805, 13.11986097179723758081529521329, 13.80135983963308699623731195459, 15.03478330793493775732830132103, 15.929476195586250603956472745229, 16.77469127774407178163839634020, 17.6044163848114487683390864506, 18.56077049243462218290112101511, 18.90684394547182400939617468787, 19.52695561870713483138991454033, 21.054166982767862530566535464983, 21.34170195281192629666387195544

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