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Properties

Label 2-31768-1.1-c1-0-4
Degree 22
Conductor 3176831768
Sign −1-1
Analytic cond. 253.668253.668
Root an. cond. 15.926915.9269
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 2·7-s + 9-s + 11-s − 5·13-s + 2·17-s − 4·21-s − 8·23-s − 5·25-s + 4·27-s + 5·29-s − 2·33-s − 2·37-s + 10·39-s + 2·41-s − 11·43-s + 9·47-s − 3·49-s − 4·51-s + 2·53-s + 14·59-s + 7·61-s + 2·63-s − 2·67-s + 16·69-s − 3·71-s − 4·73-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.755·7-s + 1/3·9-s + 0.301·11-s − 1.38·13-s + 0.485·17-s − 0.872·21-s − 1.66·23-s − 25-s + 0.769·27-s + 0.928·29-s − 0.348·33-s − 0.328·37-s + 1.60·39-s + 0.312·41-s − 1.67·43-s + 1.31·47-s − 3/7·49-s − 0.560·51-s + 0.274·53-s + 1.82·59-s + 0.896·61-s + 0.251·63-s − 0.244·67-s + 1.92·69-s − 0.356·71-s − 0.468·73-s + ⋯

Functional equation

Λ(s)=(31768s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 31768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(31768s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 31768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 3176831768    =    23⋅11⋅1922^{3} \cdot 11 \cdot 19^{2}
Sign: −1-1
Analytic conductor: 253.668253.668
Root analytic conductor: 15.926915.9269
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 31768, ( :1/2), −1)(2,\ 31768,\ (\ :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
11 1−T 1 - T
19 1 1
good3 1+2T+pT2 1 + 2 T + p T^{2}
5 1+pT2 1 + p T^{2}
7 1−2T+pT2 1 - 2 T + p T^{2}
13 1+5T+pT2 1 + 5 T + p T^{2}
17 1−2T+pT2 1 - 2 T + p T^{2}
23 1+8T+pT2 1 + 8 T + p T^{2}
29 1−5T+pT2 1 - 5 T + p T^{2}
31 1+pT2 1 + p T^{2}
37 1+2T+pT2 1 + 2 T + p T^{2}
41 1−2T+pT2 1 - 2 T + p T^{2}
43 1+11T+pT2 1 + 11 T + p T^{2}
47 1−9T+pT2 1 - 9 T + p T^{2}
53 1−2T+pT2 1 - 2 T + p T^{2}
59 1−14T+pT2 1 - 14 T + p T^{2}
61 1−7T+pT2 1 - 7 T + p T^{2}
67 1+2T+pT2 1 + 2 T + p T^{2}
71 1+3T+pT2 1 + 3 T + p T^{2}
73 1+4T+pT2 1 + 4 T + p T^{2}
79 1+pT2 1 + p T^{2}
83 1+17T+pT2 1 + 17 T + p T^{2}
89 1−7T+pT2 1 - 7 T + p T^{2}
97 1−7T+pT2 1 - 7 T + p 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

−15.44328743795491, −14.66780246370801, −14.28467532004677, −13.95899174043508, −13.09137375968877, −12.47122866599706, −11.93930235042538, −11.67767315089525, −11.36143783163376, −10.39120281610249, −10.06821835599959, −9.760502290346871, −8.614962127304729, −8.389144990489165, −7.452744084843014, −7.244363498978471, −6.277068346021759, −5.945262292861671, −5.236741388238639, −4.821703110235150, −4.210242662804741, −3.435039894087968, −2.406233885089577, −1.847287665696157, −0.8362709909396113, 0, 0.8362709909396113, 1.847287665696157, 2.406233885089577, 3.435039894087968, 4.210242662804741, 4.821703110235150, 5.236741388238639, 5.945262292861671, 6.277068346021759, 7.244363498978471, 7.452744084843014, 8.389144990489165, 8.614962127304729, 9.760502290346871, 10.06821835599959, 10.39120281610249, 11.36143783163376, 11.67767315089525, 11.93930235042538, 12.47122866599706, 13.09137375968877, 13.95899174043508, 14.28467532004677, 14.66780246370801, 15.44328743795491

Graph of the ZZ-function along the critical line