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Properties

Label 2-475-19.18-c0-0-1
Degree 22
Conductor 475475
Sign 11
Analytic cond. 0.2370550.237055
Root an. cond. 0.4868830.486883
Motivic weight 00
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

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

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

Invariants

Degree: 22
Conductor: 475475    =    52⋅195^{2} \cdot 19
Sign: 11
Analytic conductor: 0.2370550.237055
Root analytic conductor: 0.4868830.486883
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: χ475(151,⋅)\chi_{475} (151, \cdot )
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 475, ( :0), 1)(2,\ 475,\ (\ :0),\ 1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.0258561141.025856114
L(12)L(\frac12) ≈\approx 1.0258561141.025856114
L(1)L(1) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad5 1 1
19 1+T 1 + T
good2 (1−T)(1+T) ( 1 - T )( 1 + T )
3 (1−T)(1+T) ( 1 - T )( 1 + T )
7 1+T2 1 + T^{2}
11 (1+T)2 ( 1 + T )^{2}
13 (1−T)(1+T) ( 1 - T )( 1 + T )
17 1+T2 1 + T^{2}
23 1+T2 1 + T^{2}
29 (1−T)(1+T) ( 1 - T )( 1 + T )
31 (1−T)(1+T) ( 1 - T )( 1 + T )
37 (1−T)(1+T) ( 1 - T )( 1 + T )
41 (1−T)(1+T) ( 1 - T )( 1 + T )
43 1+T2 1 + T^{2}
47 1+T2 1 + T^{2}
53 (1−T)(1+T) ( 1 - T )( 1 + T )
59 (1−T)(1+T) ( 1 - T )( 1 + T )
61 (1+T)2 ( 1 + T )^{2}
67 (1−T)(1+T) ( 1 - T )( 1 + T )
71 (1−T)(1+T) ( 1 - T )( 1 + T )
73 1+T2 1 + T^{2}
79 (1−T)(1+T) ( 1 - T )( 1 + T )
83 1+T2 1 + T^{2}
89 (1−T)(1+T) ( 1 - T )( 1 + T )
97 (1−T)(1+T) ( 1 - T )( 1 + T )
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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 ZZ-function along the critical line