Properties

Label 2-475-475.189-c0-0-0
Degree $2$
Conductor $475$
Sign $0.187 - 0.982i$
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  + (−0.309 + 0.951i)4-s + (−0.809 − 0.587i)5-s + 1.17i·7-s + (0.809 + 0.587i)9-s + (−1.30 + 0.951i)11-s + (−0.809 − 0.587i)16-s + (1.80 − 0.587i)17-s + (0.309 + 0.951i)19-s + (0.809 − 0.587i)20-s + (−0.690 − 0.951i)23-s + (0.309 + 0.951i)25-s + (−1.11 − 0.363i)28-s + (0.690 − 0.951i)35-s + (−0.809 + 0.587i)36-s − 1.90i·43-s + (−0.499 − 1.53i)44-s + ⋯
L(s)  = 1  + (−0.309 + 0.951i)4-s + (−0.809 − 0.587i)5-s + 1.17i·7-s + (0.809 + 0.587i)9-s + (−1.30 + 0.951i)11-s + (−0.809 − 0.587i)16-s + (1.80 − 0.587i)17-s + (0.309 + 0.951i)19-s + (0.809 − 0.587i)20-s + (−0.690 − 0.951i)23-s + (0.309 + 0.951i)25-s + (−1.11 − 0.363i)28-s + (0.690 − 0.951i)35-s + (−0.809 + 0.587i)36-s − 1.90i·43-s + (−0.499 − 1.53i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.187 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.187 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.187 - 0.982i$
Analytic conductor: \(0.237055\)
Root analytic conductor: \(0.486883\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (189, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :0),\ 0.187 - 0.982i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6948572259\)
\(L(\frac12)\) \(\approx\) \(0.6948572259\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.809 + 0.587i)T \)
19 \( 1 + (-0.309 - 0.951i)T \)
good2 \( 1 + (0.309 - 0.951i)T^{2} \)
3 \( 1 + (-0.809 - 0.587i)T^{2} \)
7 \( 1 - 1.17iT - T^{2} \)
11 \( 1 + (1.30 - 0.951i)T + (0.309 - 0.951i)T^{2} \)
13 \( 1 + (0.309 + 0.951i)T^{2} \)
17 \( 1 + (-1.80 + 0.587i)T + (0.809 - 0.587i)T^{2} \)
23 \( 1 + (0.690 + 0.951i)T + (-0.309 + 0.951i)T^{2} \)
29 \( 1 + (0.809 + 0.587i)T^{2} \)
31 \( 1 + (0.809 - 0.587i)T^{2} \)
37 \( 1 + (0.309 + 0.951i)T^{2} \)
41 \( 1 + (-0.309 - 0.951i)T^{2} \)
43 \( 1 + 1.90iT - T^{2} \)
47 \( 1 + (0.809 + 0.587i)T^{2} \)
53 \( 1 + (-0.809 - 0.587i)T^{2} \)
59 \( 1 + (-0.309 - 0.951i)T^{2} \)
61 \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \)
67 \( 1 + (-0.809 + 0.587i)T^{2} \)
71 \( 1 + (0.809 + 0.587i)T^{2} \)
73 \( 1 + (-0.309 + 0.951i)T^{2} \)
79 \( 1 + (0.809 + 0.587i)T^{2} \)
83 \( 1 + (-1.80 + 0.587i)T + (0.809 - 0.587i)T^{2} \)
89 \( 1 + (-0.309 + 0.951i)T^{2} \)
97 \( 1 + (-0.809 - 0.587i)T^{2} \)
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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

−11.87933652198433878513471069887, −10.40907209332789332122411361977, −9.608214347272229386734718227830, −8.477187151466016427696281918093, −7.83727882344111302115200046105, −7.31725336482532786333466471352, −5.44531631168305818233522189093, −4.77961422110777909717812672801, −3.60526783870640191779209270231, −2.30564430437665130897188338848, 0.978776212574264106031507618886, 3.22470335007746915705417134622, 4.15242105935830573817782920367, 5.34524417964202535976329559971, 6.41691272283075467280982156779, 7.47340117282881315843328996968, 8.059135648538837434917684708481, 9.557536894224856921804389280368, 10.28266028916752015184166060724, 10.78612107398408201846071956577

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