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Properties

Label 2-475-1.1-c1-0-27
Degree 22
Conductor 475475
Sign −1-1
Analytic cond. 3.792893.79289
Root an. cond. 1.947531.94753
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.48·2-s − 0.806·3-s + 0.193·4-s − 1.19·6-s − 3.35·7-s − 2.67·8-s − 2.35·9-s + 0.962·11-s − 0.156·12-s − 6.15·13-s − 4.96·14-s − 4.35·16-s + 6.31·17-s − 3.48·18-s − 19-s + 2.70·21-s + 1.42·22-s + 4.96·23-s + 2.15·24-s − 9.11·26-s + 4.31·27-s − 0.649·28-s − 3.61·29-s − 5.92·31-s − 1.09·32-s − 0.775·33-s + 9.35·34-s + ⋯
L(s)  = 1  + 1.04·2-s − 0.465·3-s + 0.0969·4-s − 0.487·6-s − 1.26·7-s − 0.945·8-s − 0.783·9-s + 0.290·11-s − 0.0451·12-s − 1.70·13-s − 1.32·14-s − 1.08·16-s + 1.53·17-s − 0.820·18-s − 0.229·19-s + 0.589·21-s + 0.303·22-s + 1.03·23-s + 0.440·24-s − 1.78·26-s + 0.829·27-s − 0.122·28-s − 0.670·29-s − 1.06·31-s − 0.193·32-s − 0.135·33-s + 1.60·34-s + ⋯

Functional equation

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

Invariants

Degree: 22
Conductor: 475475    =    52⋅195^{2} \cdot 19
Sign: −1-1
Analytic conductor: 3.792893.79289
Root analytic conductor: 1.947531.94753
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 475, ( :1/2), −1)(2,\ 475,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
L(32)L(\frac{3}{2}) 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 1−1.48T+2T2 1 - 1.48T + 2T^{2}
3 1+0.806T+3T2 1 + 0.806T + 3T^{2}
7 1+3.35T+7T2 1 + 3.35T + 7T^{2}
11 1−0.962T+11T2 1 - 0.962T + 11T^{2}
13 1+6.15T+13T2 1 + 6.15T + 13T^{2}
17 1−6.31T+17T2 1 - 6.31T + 17T^{2}
23 1−4.96T+23T2 1 - 4.96T + 23T^{2}
29 1+3.61T+29T2 1 + 3.61T + 29T^{2}
31 1+5.92T+31T2 1 + 5.92T + 31T^{2}
37 1+10.1T+37T2 1 + 10.1T + 37T^{2}
41 1−6.31T+41T2 1 - 6.31T + 41T^{2}
43 1−4.12T+43T2 1 - 4.12T + 43T^{2}
47 1+3.35T+47T2 1 + 3.35T + 47T^{2}
53 1+1.84T+53T2 1 + 1.84T + 53T^{2}
59 1+6.38T+59T2 1 + 6.38T + 59T^{2}
61 1+11.2T+61T2 1 + 11.2T + 61T^{2}
67 1−6.73T+67T2 1 - 6.73T + 67T^{2}
71 1+0.775T+71T2 1 + 0.775T + 71T^{2}
73 1+0.387T+73T2 1 + 0.387T + 73T^{2}
79 1+0.836T+79T2 1 + 0.836T + 79T^{2}
83 1−7.03T+83T2 1 - 7.03T + 83T^{2}
89 1−7.08T+89T2 1 - 7.08T + 89T^{2}
97 1+10.9T+97T2 1 + 10.9T + 97T^{2}
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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

−10.68706585680744218921860629453, −9.596030474136190821664917785490, −9.066418741591584822201264381691, −7.52786815767506675418148855799, −6.53381958851570359278476920409, −5.61833942145190438759590697906, −4.99171169998799150443949090789, −3.58462042181120631379624837386, −2.81151268124156327008609849566, 0, 2.81151268124156327008609849566, 3.58462042181120631379624837386, 4.99171169998799150443949090789, 5.61833942145190438759590697906, 6.53381958851570359278476920409, 7.52786815767506675418148855799, 9.066418741591584822201264381691, 9.596030474136190821664917785490, 10.68706585680744218921860629453

Graph of the ZZ-function along the critical line