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Properties

Label 2-7488-1.1-c1-0-70
Degree 22
Conductor 74887488
Sign −1-1
Analytic cond. 59.791959.7919
Root an. cond. 7.732527.73252
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  − 2·5-s − 2·7-s + 4·11-s + 13-s − 2·19-s − 4·23-s − 25-s − 2·31-s + 4·35-s + 10·37-s + 2·41-s + 8·43-s − 3·49-s − 12·53-s − 8·55-s + 12·59-s + 6·61-s − 2·65-s − 6·67-s + 8·71-s − 2·73-s − 8·77-s − 12·79-s − 4·83-s + 14·89-s − 2·91-s + 4·95-s + ⋯
L(s)  = 1  − 0.894·5-s − 0.755·7-s + 1.20·11-s + 0.277·13-s − 0.458·19-s − 0.834·23-s − 1/5·25-s − 0.359·31-s + 0.676·35-s + 1.64·37-s + 0.312·41-s + 1.21·43-s − 3/7·49-s − 1.64·53-s − 1.07·55-s + 1.56·59-s + 0.768·61-s − 0.248·65-s − 0.733·67-s + 0.949·71-s − 0.234·73-s − 0.911·77-s − 1.35·79-s − 0.439·83-s + 1.48·89-s − 0.209·91-s + 0.410·95-s + ⋯

Functional equation

Λ(s)=(7488s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 7488 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(7488s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 7488 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 74887488    =    26⋅32⋅132^{6} \cdot 3^{2} \cdot 13
Sign: −1-1
Analytic conductor: 59.791959.7919
Root analytic conductor: 7.732527.73252
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 7488, ( :1/2), −1)(2,\ 7488,\ (\ :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
3 1 1
13 1−T 1 - T
good5 1+2T+pT2 1 + 2 T + p T^{2}
7 1+2T+pT2 1 + 2 T + p T^{2}
11 1−4T+pT2 1 - 4 T + p T^{2}
17 1+pT2 1 + p T^{2}
19 1+2T+pT2 1 + 2 T + p T^{2}
23 1+4T+pT2 1 + 4 T + p T^{2}
29 1+pT2 1 + p T^{2}
31 1+2T+pT2 1 + 2 T + p T^{2}
37 1−10T+pT2 1 - 10 T + p T^{2}
41 1−2T+pT2 1 - 2 T + p T^{2}
43 1−8T+pT2 1 - 8 T + p T^{2}
47 1+pT2 1 + p T^{2}
53 1+12T+pT2 1 + 12 T + p T^{2}
59 1−12T+pT2 1 - 12 T + p T^{2}
61 1−6T+pT2 1 - 6 T + p T^{2}
67 1+6T+pT2 1 + 6 T + p T^{2}
71 1−8T+pT2 1 - 8 T + p T^{2}
73 1+2T+pT2 1 + 2 T + p T^{2}
79 1+12T+pT2 1 + 12 T + p T^{2}
83 1+4T+pT2 1 + 4 T + p T^{2}
89 1−14T+pT2 1 - 14 T + p T^{2}
97 1+10T+pT2 1 + 10 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

−7.64979557871995762400784333728, −6.73592272653504112856858596335, −6.29882762515940384651380774168, −5.58334981310967040664323173402, −4.34375032280870050600303274142, −4.01978688582189601994726957884, −3.30845303295264910359826466552, −2.31483949994290673709373180139, −1.13992312389026331214163706283, 0, 1.13992312389026331214163706283, 2.31483949994290673709373180139, 3.30845303295264910359826466552, 4.01978688582189601994726957884, 4.34375032280870050600303274142, 5.58334981310967040664323173402, 6.29882762515940384651380774168, 6.73592272653504112856858596335, 7.64979557871995762400784333728

Graph of the ZZ-function along the critical line