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Properties

Label 2-9408-1.1-c1-0-156
Degree 22
Conductor 94089408
Sign −1-1
Analytic cond. 75.123275.1232
Root an. cond. 8.667368.66736
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  + 3-s + 3·5-s + 9-s − 3·11-s − 4·13-s + 3·15-s − 4·19-s + 4·25-s + 27-s − 9·29-s + 31-s − 3·33-s − 8·37-s − 4·39-s + 10·43-s + 3·45-s + 6·47-s + 3·53-s − 9·55-s − 4·57-s + 3·59-s − 10·61-s − 12·65-s + 10·67-s − 6·71-s − 2·73-s + 4·75-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.34·5-s + 1/3·9-s − 0.904·11-s − 1.10·13-s + 0.774·15-s − 0.917·19-s + 4/5·25-s + 0.192·27-s − 1.67·29-s + 0.179·31-s − 0.522·33-s − 1.31·37-s − 0.640·39-s + 1.52·43-s + 0.447·45-s + 0.875·47-s + 0.412·53-s − 1.21·55-s − 0.529·57-s + 0.390·59-s − 1.28·61-s − 1.48·65-s + 1.22·67-s − 0.712·71-s − 0.234·73-s + 0.461·75-s + ⋯

Functional equation

Λ(s)=(9408s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(9408s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 94089408    =    26⋅3⋅722^{6} \cdot 3 \cdot 7^{2}
Sign: −1-1
Analytic conductor: 75.123275.1232
Root analytic conductor: 8.667368.66736
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 9408, ( :1/2), −1)(2,\ 9408,\ (\ :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−T 1 - T
7 1 1
good5 1−3T+pT2 1 - 3 T + p T^{2}
11 1+3T+pT2 1 + 3 T + p T^{2}
13 1+4T+pT2 1 + 4 T + p T^{2}
17 1+pT2 1 + p T^{2}
19 1+4T+pT2 1 + 4 T + p T^{2}
23 1+pT2 1 + p T^{2}
29 1+9T+pT2 1 + 9 T + p T^{2}
31 1−T+pT2 1 - T + p T^{2}
37 1+8T+pT2 1 + 8 T + p T^{2}
41 1+pT2 1 + p T^{2}
43 1−10T+pT2 1 - 10 T + p T^{2}
47 1−6T+pT2 1 - 6 T + p T^{2}
53 1−3T+pT2 1 - 3 T + p T^{2}
59 1−3T+pT2 1 - 3 T + p T^{2}
61 1+10T+pT2 1 + 10 T + p T^{2}
67 1−10T+pT2 1 - 10 T + p T^{2}
71 1+6T+pT2 1 + 6 T + p T^{2}
73 1+2T+pT2 1 + 2 T + p T^{2}
79 1+T+pT2 1 + T + p T^{2}
83 1+9T+pT2 1 + 9 T + p T^{2}
89 1+6T+pT2 1 + 6 T + p T^{2}
97 1−T+pT2 1 - 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

−7.33394397090324092400049797646, −6.80616770693985958913865603865, −5.78142581425042930916248854062, −5.48030169770383048419949507799, −4.63005992613826729096856469814, −3.80867461709981690563059575879, −2.67612385219831355074975282085, −2.32908448991463422849641030373, −1.54615518523511024726550509475, 0, 1.54615518523511024726550509475, 2.32908448991463422849641030373, 2.67612385219831355074975282085, 3.80867461709981690563059575879, 4.63005992613826729096856469814, 5.48030169770383048419949507799, 5.78142581425042930916248854062, 6.80616770693985958913865603865, 7.33394397090324092400049797646

Graph of the ZZ-function along the critical line