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Properties

Label 2-475-19.18-c2-0-0
Degree $2$
Conductor $475$
Sign $0.205 - 0.978i$
Analytic cond. $12.9428$
Root an. cond. $3.59761$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.79i·2-s + 1.15i·3-s − 3.82·4-s + 3.24·6-s − 6.64·7-s − 0.480i·8-s + 7.65·9-s − 6.24·11-s − 4.43i·12-s + 18.6i·13-s + 18.5i·14-s − 16.6·16-s − 29.3·17-s − 21.4i·18-s + (3.89 − 18.5i)19-s + ⋯
L(s)  = 1  − 1.39i·2-s + 0.386i·3-s − 0.957·4-s + 0.540·6-s − 0.949·7-s − 0.0600i·8-s + 0.850·9-s − 0.567·11-s − 0.369i·12-s + 1.43i·13-s + 1.32i·14-s − 1.04·16-s − 1.72·17-s − 1.19i·18-s + (0.205 − 0.978i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.205 - 0.978i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.205 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.205 - 0.978i$
Analytic conductor: \(12.9428\)
Root analytic conductor: \(3.59761\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1),\ 0.205 - 0.978i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.2594768727\)
\(L(\frac12)\) \(\approx\) \(0.2594768727\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 + (-3.89 + 18.5i)T \)
good2 \( 1 + 2.79iT - 4T^{2} \)
3 \( 1 - 1.15iT - 9T^{2} \)
7 \( 1 + 6.64T + 49T^{2} \)
11 \( 1 + 6.24T + 121T^{2} \)
13 \( 1 - 18.6iT - 169T^{2} \)
17 \( 1 + 29.3T + 289T^{2} \)
23 \( 1 + 19.9T + 529T^{2} \)
29 \( 1 - 29.4iT - 841T^{2} \)
31 \( 1 - 26.2iT - 961T^{2} \)
37 \( 1 + 47.4iT - 1.36e3T^{2} \)
41 \( 1 - 41.7iT - 1.68e3T^{2} \)
43 \( 1 + 14.9T + 1.84e3T^{2} \)
47 \( 1 + 5.50T + 2.20e3T^{2} \)
53 \( 1 - 55.4iT - 2.80e3T^{2} \)
59 \( 1 - 86.5iT - 3.48e3T^{2} \)
61 \( 1 - 20.5T + 3.72e3T^{2} \)
67 \( 1 - 0.363iT - 4.48e3T^{2} \)
71 \( 1 + 100. iT - 5.04e3T^{2} \)
73 \( 1 + 0.472T + 5.32e3T^{2} \)
79 \( 1 + 120. iT - 6.24e3T^{2} \)
83 \( 1 + 42.1T + 6.88e3T^{2} \)
89 \( 1 - 4.51iT - 7.92e3T^{2} \)
97 \( 1 + 11.5iT - 9.40e3T^{2} \)
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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

−10.89422080004862009961873493747, −10.30020450480713864717697345323, −9.319154879817651748750587295405, −9.003634803245273235653959551953, −7.13755453438823124616954301540, −6.51193551544483362663190692435, −4.71097314664842030563519724167, −4.04191243156398383499037078850, −2.86617409404588863934570840527, −1.75120735906533236319529274264, 0.098719838110280187201087057147, 2.35245850151161886471685521586, 3.97015830640740308761687586498, 5.23886815393178327290864066844, 6.21058089727886748621270441150, 6.79151731253890342402531309684, 7.82836103977858310730277529999, 8.299278825164217175369094674507, 9.652550438149398376009241179912, 10.30250319169182706289077682941

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