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Properties

Label 2-475-475.379-c0-0-0
Degree $2$
Conductor $475$
Sign $0.968 - 0.248i$
Analytic cond. $0.237055$
Root an. cond. $0.486883$
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)4-s + (0.309 + 0.951i)5-s − 1.90i·7-s + (−0.309 − 0.951i)9-s + (−0.190 + 0.587i)11-s + (0.309 + 0.951i)16-s + (0.690 + 0.951i)17-s + (−0.809 + 0.587i)19-s + (−0.309 + 0.951i)20-s + (−1.80 − 0.587i)23-s + (−0.809 + 0.587i)25-s + (1.11 − 1.53i)28-s + (1.80 − 0.587i)35-s + (0.309 − 0.951i)36-s − 1.17i·43-s + (−0.5 + 0.363i)44-s + ⋯
L(s)  = 1  + (0.809 + 0.587i)4-s + (0.309 + 0.951i)5-s − 1.90i·7-s + (−0.309 − 0.951i)9-s + (−0.190 + 0.587i)11-s + (0.309 + 0.951i)16-s + (0.690 + 0.951i)17-s + (−0.809 + 0.587i)19-s + (−0.309 + 0.951i)20-s + (−1.80 − 0.587i)23-s + (−0.809 + 0.587i)25-s + (1.11 − 1.53i)28-s + (1.80 − 0.587i)35-s + (0.309 − 0.951i)36-s − 1.17i·43-s + (−0.5 + 0.363i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.968 - 0.248i$
Analytic conductor: \(0.237055\)
Root analytic conductor: \(0.486883\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :0),\ 0.968 - 0.248i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.020540789\)
\(L(\frac12)\) \(\approx\) \(1.020540789\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.309 - 0.951i)T \)
19 \( 1 + (0.809 - 0.587i)T \)
good2 \( 1 + (-0.809 - 0.587i)T^{2} \)
3 \( 1 + (0.309 + 0.951i)T^{2} \)
7 \( 1 + 1.90iT - T^{2} \)
11 \( 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2} \)
13 \( 1 + (-0.809 + 0.587i)T^{2} \)
17 \( 1 + (-0.690 - 0.951i)T + (-0.309 + 0.951i)T^{2} \)
23 \( 1 + (1.80 + 0.587i)T + (0.809 + 0.587i)T^{2} \)
29 \( 1 + (-0.309 - 0.951i)T^{2} \)
31 \( 1 + (-0.309 + 0.951i)T^{2} \)
37 \( 1 + (-0.809 + 0.587i)T^{2} \)
41 \( 1 + (0.809 - 0.587i)T^{2} \)
43 \( 1 + 1.17iT - T^{2} \)
47 \( 1 + (-0.309 - 0.951i)T^{2} \)
53 \( 1 + (0.309 + 0.951i)T^{2} \)
59 \( 1 + (0.809 - 0.587i)T^{2} \)
61 \( 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2} \)
67 \( 1 + (0.309 - 0.951i)T^{2} \)
71 \( 1 + (-0.309 - 0.951i)T^{2} \)
73 \( 1 + (0.809 + 0.587i)T^{2} \)
79 \( 1 + (-0.309 - 0.951i)T^{2} \)
83 \( 1 + (-0.690 - 0.951i)T + (-0.309 + 0.951i)T^{2} \)
89 \( 1 + (0.809 + 0.587i)T^{2} \)
97 \( 1 + (0.309 + 0.951i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.08564531258956489912086284274, −10.34626425106951248417943597613, −9.962491290393719875652342363670, −8.220554886297140339835032960158, −7.51818624697035741913515575625, −6.67256833850442352570675537296, −6.10710462215372443126127097996, −4.00343695033569974392658940336, −3.53146896102969444768753435180, −1.96735852384102586528512784752, 1.93112197225081561036352910249, 2.74203902942505081087523692548, 4.89924737773433295499930340551, 5.64023156466830927648206487152, 6.12224877423187121168136150023, 7.75546896515520546100149609462, 8.534048310231528204789974487993, 9.382467312482019860847898996729, 10.23193411506540867269356057908, 11.47442423723914685018147300023

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