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Properties

Label 2-475-19.18-c0-0-1
Degree $2$
Conductor $475$
Sign $1$
Analytic cond. $0.237055$
Root an. cond. $0.486883$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4-s + 9-s − 2·11-s + 16-s − 19-s + 36-s − 2·44-s − 49-s − 2·61-s + 64-s − 76-s + 81-s − 2·99-s − 2·101-s + ⋯
L(s)  = 1  + 4-s + 9-s − 2·11-s + 16-s − 19-s + 36-s − 2·44-s − 49-s − 2·61-s + 64-s − 76-s + 81-s − 2·99-s − 2·101-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $1$
Analytic conductor: \(0.237055\)
Root analytic conductor: \(0.486883\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{475} (151, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.025856114\)
\(L(\frac12)\) \(\approx\) \(1.025856114\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 + T \)
good2 \( ( 1 - T )( 1 + T ) \)
3 \( ( 1 - T )( 1 + T ) \)
7 \( 1 + T^{2} \)
11 \( ( 1 + T )^{2} \)
13 \( ( 1 - T )( 1 + T ) \)
17 \( 1 + T^{2} \)
23 \( 1 + T^{2} \)
29 \( ( 1 - T )( 1 + T ) \)
31 \( ( 1 - T )( 1 + T ) \)
37 \( ( 1 - T )( 1 + T ) \)
41 \( ( 1 - T )( 1 + T ) \)
43 \( 1 + T^{2} \)
47 \( 1 + T^{2} \)
53 \( ( 1 - T )( 1 + T ) \)
59 \( ( 1 - T )( 1 + T ) \)
61 \( ( 1 + T )^{2} \)
67 \( ( 1 - T )( 1 + T ) \)
71 \( ( 1 - T )( 1 + T ) \)
73 \( 1 + T^{2} \)
79 \( ( 1 - T )( 1 + T ) \)
83 \( 1 + T^{2} \)
89 \( ( 1 - T )( 1 + T ) \)
97 \( ( 1 - T )( 1 + T ) \)
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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

−10.92589093376341656534887151392, −10.55255274398139820172084891263, −9.705610040248617143663172232154, −8.238351875950604978574364954925, −7.58302198357719280831316953677, −6.70986731944779184213579089094, −5.65739929328088967394736940749, −4.55107441902203862755643435150, −3.04174701047875108100977709927, −1.94756753976557909188690809793, 1.94756753976557909188690809793, 3.04174701047875108100977709927, 4.55107441902203862755643435150, 5.65739929328088967394736940749, 6.70986731944779184213579089094, 7.58302198357719280831316953677, 8.238351875950604978574364954925, 9.705610040248617143663172232154, 10.55255274398139820172084891263, 10.92589093376341656534887151392

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