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Properties

Label 2-3800-1.1-c1-0-64
Degree 22
Conductor 38003800
Sign −1-1
Analytic cond. 30.343130.3431
Root an. cond. 5.508465.50846
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  + 0.347·3-s − 3.41·7-s − 2.87·9-s + 2.41·11-s + 6.29·13-s − 2.34·17-s + 19-s − 1.18·21-s + 2.49·23-s − 2.04·27-s − 8.17·29-s − 2.77·31-s + 0.837·33-s − 0.977·37-s + 2.18·39-s − 3.49·41-s − 2.75·43-s + 6.29·47-s + 4.63·49-s − 0.815·51-s + 2.38·53-s + 0.347·57-s + 3.67·59-s − 12.7·61-s + 9.82·63-s + 2.41·67-s + 0.864·69-s + ⋯
L(s)  = 1  + 0.200·3-s − 1.28·7-s − 0.959·9-s + 0.727·11-s + 1.74·13-s − 0.569·17-s + 0.229·19-s − 0.258·21-s + 0.519·23-s − 0.392·27-s − 1.51·29-s − 0.498·31-s + 0.145·33-s − 0.160·37-s + 0.349·39-s − 0.545·41-s − 0.420·43-s + 0.917·47-s + 0.662·49-s − 0.114·51-s + 0.328·53-s + 0.0460·57-s + 0.478·59-s − 1.63·61-s + 1.23·63-s + 0.294·67-s + 0.104·69-s + ⋯

Functional equation

Λ(s)=(3800s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(3800s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 38003800    =    23⋅52⋅192^{3} \cdot 5^{2} \cdot 19
Sign: −1-1
Analytic conductor: 30.343130.3431
Root analytic conductor: 5.508465.50846
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 3800, ( :1/2), −1)(2,\ 3800,\ (\ :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)
bad2 1 1
5 1 1
19 1−T 1 - T
good3 1−0.347T+3T2 1 - 0.347T + 3T^{2}
7 1+3.41T+7T2 1 + 3.41T + 7T^{2}
11 1−2.41T+11T2 1 - 2.41T + 11T^{2}
13 1−6.29T+13T2 1 - 6.29T + 13T^{2}
17 1+2.34T+17T2 1 + 2.34T + 17T^{2}
23 1−2.49T+23T2 1 - 2.49T + 23T^{2}
29 1+8.17T+29T2 1 + 8.17T + 29T^{2}
31 1+2.77T+31T2 1 + 2.77T + 31T^{2}
37 1+0.977T+37T2 1 + 0.977T + 37T^{2}
41 1+3.49T+41T2 1 + 3.49T + 41T^{2}
43 1+2.75T+43T2 1 + 2.75T + 43T^{2}
47 1−6.29T+47T2 1 - 6.29T + 47T^{2}
53 1−2.38T+53T2 1 - 2.38T + 53T^{2}
59 1−3.67T+59T2 1 - 3.67T + 59T^{2}
61 1+12.7T+61T2 1 + 12.7T + 61T^{2}
67 1−2.41T+67T2 1 - 2.41T + 67T^{2}
71 1−4.51T+71T2 1 - 4.51T + 71T^{2}
73 1−1.81T+73T2 1 - 1.81T + 73T^{2}
79 1+5.04T+79T2 1 + 5.04T + 79T^{2}
83 1+8.07T+83T2 1 + 8.07T + 83T^{2}
89 1+2.94T+89T2 1 + 2.94T + 89T^{2}
97 1+3.09T+97T2 1 + 3.09T + 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

−8.326557700732181785331186957112, −7.30515312906735016620445692058, −6.51812414607771200458582306400, −6.03976435627965018324782052911, −5.31751480890423155789919971311, −3.90131940486388605497069420213, −3.55421250105112568367958548171, −2.66880638982154988135143913954, −1.40581723816416527127611914855, 0, 1.40581723816416527127611914855, 2.66880638982154988135143913954, 3.55421250105112568367958548171, 3.90131940486388605497069420213, 5.31751480890423155789919971311, 6.03976435627965018324782052911, 6.51812414607771200458582306400, 7.30515312906735016620445692058, 8.326557700732181785331186957112

Graph of the ZZ-function along the critical line