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Properties

Label 2-396-1.1-c3-0-11
Degree 22
Conductor 396396
Sign −1-1
Analytic cond. 23.364723.3647
Root an. cond. 4.833714.83371
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 7·5-s − 26·7-s + 11·11-s + 52·13-s − 46·17-s − 96·19-s − 27·23-s − 76·25-s − 16·29-s − 293·31-s − 182·35-s − 29·37-s + 472·41-s − 110·43-s + 224·47-s + 333·49-s − 754·53-s + 77·55-s − 825·59-s − 548·61-s + 364·65-s − 123·67-s − 1.00e3·71-s − 1.02e3·73-s − 286·77-s + 526·79-s + 158·83-s + ⋯
L(s)  = 1  + 0.626·5-s − 1.40·7-s + 0.301·11-s + 1.10·13-s − 0.656·17-s − 1.15·19-s − 0.244·23-s − 0.607·25-s − 0.102·29-s − 1.69·31-s − 0.878·35-s − 0.128·37-s + 1.79·41-s − 0.390·43-s + 0.695·47-s + 0.970·49-s − 1.95·53-s + 0.188·55-s − 1.82·59-s − 1.15·61-s + 0.694·65-s − 0.224·67-s − 1.67·71-s − 1.63·73-s − 0.423·77-s + 0.749·79-s + 0.208·83-s + ⋯

Functional equation

Λ(s)=(396s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 396 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(396s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 396 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 396396    =    22⋅32⋅112^{2} \cdot 3^{2} \cdot 11
Sign: −1-1
Analytic conductor: 23.364723.3647
Root analytic conductor: 4.833714.83371
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 396, ( :3/2), −1)(2,\ 396,\ (\ :3/2),\ -1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
L(52)L(\frac{5}{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
3 1 1
11 1−pT 1 - p T
good5 1−7T+p3T2 1 - 7 T + p^{3} T^{2}
7 1+26T+p3T2 1 + 26 T + p^{3} T^{2}
13 1−4pT+p3T2 1 - 4 p T + p^{3} T^{2}
17 1+46T+p3T2 1 + 46 T + p^{3} T^{2}
19 1+96T+p3T2 1 + 96 T + p^{3} T^{2}
23 1+27T+p3T2 1 + 27 T + p^{3} T^{2}
29 1+16T+p3T2 1 + 16 T + p^{3} T^{2}
31 1+293T+p3T2 1 + 293 T + p^{3} T^{2}
37 1+29T+p3T2 1 + 29 T + p^{3} T^{2}
41 1−472T+p3T2 1 - 472 T + p^{3} T^{2}
43 1+110T+p3T2 1 + 110 T + p^{3} T^{2}
47 1−224T+p3T2 1 - 224 T + p^{3} T^{2}
53 1+754T+p3T2 1 + 754 T + p^{3} T^{2}
59 1+825T+p3T2 1 + 825 T + p^{3} T^{2}
61 1+548T+p3T2 1 + 548 T + p^{3} T^{2}
67 1+123T+p3T2 1 + 123 T + p^{3} T^{2}
71 1+1001T+p3T2 1 + 1001 T + p^{3} T^{2}
73 1+1020T+p3T2 1 + 1020 T + p^{3} T^{2}
79 1−526T+p3T2 1 - 526 T + p^{3} T^{2}
83 1−158T+p3T2 1 - 158 T + p^{3} T^{2}
89 1−1217T+p3T2 1 - 1217 T + p^{3} T^{2}
97 1+263T+p3T2 1 + 263 T + p^{3} 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

−10.43490258510042393693729409379, −9.340391455400642175275609699197, −8.926094348847763044694111436525, −7.50570209137514883352468654595, −6.21850071225756716797766263570, −6.05456491027802811356902335353, −4.28704553877771264587731307061, −3.23370225364755618440609752171, −1.83203510615444164184116545392, 0, 1.83203510615444164184116545392, 3.23370225364755618440609752171, 4.28704553877771264587731307061, 6.05456491027802811356902335353, 6.21850071225756716797766263570, 7.50570209137514883352468654595, 8.926094348847763044694111436525, 9.340391455400642175275609699197, 10.43490258510042393693729409379

Graph of the ZZ-function along the critical line