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Properties

Label 2-475-1.1-c5-0-80
Degree 22
Conductor 475475
Sign 11
Analytic cond. 76.182376.1823
Root an. cond. 8.728248.72824
Motivic weight 55
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 7·2-s + 11·3-s + 17·4-s + 77·6-s + 197·7-s − 105·8-s − 122·9-s − 468·11-s + 187·12-s + 921·13-s + 1.37e3·14-s − 1.27e3·16-s + 1.10e3·17-s − 854·18-s + 361·19-s + 2.16e3·21-s − 3.27e3·22-s + 3.64e3·23-s − 1.15e3·24-s + 6.44e3·26-s − 4.01e3·27-s + 3.34e3·28-s + 7.52e3·29-s + 1.42e3·31-s − 5.59e3·32-s − 5.14e3·33-s + 7.74e3·34-s + ⋯
L(s)  = 1  + 1.23·2-s + 0.705·3-s + 0.531·4-s + 0.873·6-s + 1.51·7-s − 0.580·8-s − 0.502·9-s − 1.16·11-s + 0.374·12-s + 1.51·13-s + 1.88·14-s − 1.24·16-s + 0.929·17-s − 0.621·18-s + 0.229·19-s + 1.07·21-s − 1.44·22-s + 1.43·23-s − 0.409·24-s + 1.87·26-s − 1.05·27-s + 0.807·28-s + 1.66·29-s + 0.265·31-s − 0.965·32-s − 0.822·33-s + 1.14·34-s + ⋯

Functional equation

Λ(s)=(475s/2 ΓC(s) L(s)=( Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(475s/2 ΓC(s+5/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 475475    =    52⋅195^{2} \cdot 19
Sign: 11
Analytic conductor: 76.182376.1823
Root analytic conductor: 8.728248.72824
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 475, ( :5/2), 1)(2,\ 475,\ (\ :5/2),\ 1)

Particular Values

L(3)L(3) ≈\approx 6.0740299676.074029967
L(12)L(\frac12) ≈\approx 6.0740299676.074029967
L(72)L(\frac{7}{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−p2T 1 - p^{2} T
good2 1−7T+p5T2 1 - 7 T + p^{5} T^{2}
3 1−11T+p5T2 1 - 11 T + p^{5} T^{2}
7 1−197T+p5T2 1 - 197 T + p^{5} T^{2}
11 1+468T+p5T2 1 + 468 T + p^{5} T^{2}
13 1−921T+p5T2 1 - 921 T + p^{5} T^{2}
17 1−1107T+p5T2 1 - 1107 T + p^{5} T^{2}
23 1−3641T+p5T2 1 - 3641 T + p^{5} T^{2}
29 1−7525T+p5T2 1 - 7525 T + p^{5} T^{2}
31 1−1422T+p5T2 1 - 1422 T + p^{5} T^{2}
37 1−11282T+p5T2 1 - 11282 T + p^{5} T^{2}
41 1+678T+p5T2 1 + 678 T + p^{5} T^{2}
43 1+5974T+p5T2 1 + 5974 T + p^{5} T^{2}
47 1−11072T+p5T2 1 - 11072 T + p^{5} T^{2}
53 1−17461T+p5T2 1 - 17461 T + p^{5} T^{2}
59 1+46305T+p5T2 1 + 46305 T + p^{5} T^{2}
61 1−16292T+p5T2 1 - 16292 T + p^{5} T^{2}
67 1+36373T+p5T2 1 + 36373 T + p^{5} T^{2}
71 1+82208T+p5T2 1 + 82208 T + p^{5} T^{2}
73 1−43861T+p5T2 1 - 43861 T + p^{5} T^{2}
79 1+30130T+p5T2 1 + 30130 T + p^{5} T^{2}
83 1−91626T+p5T2 1 - 91626 T + p^{5} T^{2}
89 1−79170T+p5T2 1 - 79170 T + p^{5} T^{2}
97 1+128718T+p5T2 1 + 128718 T + p^{5} T^{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

−10.51112967558247513373588919102, −8.999277094674896054091501245289, −8.358223250387405484647754944374, −7.63490211236736879458588322466, −6.10484932976164333646195212573, −5.28128394215012048854442146875, −4.53309122392308457187306933915, −3.31946864949369114536833123650, −2.57964612064087253454405669060, −1.08321807040080984653596810933, 1.08321807040080984653596810933, 2.57964612064087253454405669060, 3.31946864949369114536833123650, 4.53309122392308457187306933915, 5.28128394215012048854442146875, 6.10484932976164333646195212573, 7.63490211236736879458588322466, 8.358223250387405484647754944374, 8.999277094674896054091501245289, 10.51112967558247513373588919102

Graph of the ZZ-function along the critical line