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Properties

Label 2-2368-1.1-c1-0-5
Degree 22
Conductor 23682368
Sign 11
Analytic cond. 18.908518.9085
Root an. cond. 4.348394.34839
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s − 4·7-s − 3·9-s − 2·13-s + 2·17-s + 2·19-s − 6·23-s − 25-s + 6·29-s − 2·31-s + 8·35-s + 37-s − 2·41-s + 2·43-s + 6·45-s − 4·47-s + 9·49-s + 6·53-s − 6·59-s + 6·61-s + 12·63-s + 4·65-s + 8·67-s − 4·71-s + 14·73-s + 2·79-s + 9·81-s + ⋯
L(s)  = 1  − 0.894·5-s − 1.51·7-s − 9-s − 0.554·13-s + 0.485·17-s + 0.458·19-s − 1.25·23-s − 1/5·25-s + 1.11·29-s − 0.359·31-s + 1.35·35-s + 0.164·37-s − 0.312·41-s + 0.304·43-s + 0.894·45-s − 0.583·47-s + 9/7·49-s + 0.824·53-s − 0.781·59-s + 0.768·61-s + 1.51·63-s + 0.496·65-s + 0.977·67-s − 0.474·71-s + 1.63·73-s + 0.225·79-s + 81-s + ⋯

Functional equation

Λ(s)=(2368s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(2368s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 23682368    =    26⋅372^{6} \cdot 37
Sign: 11
Analytic conductor: 18.908518.9085
Root analytic conductor: 4.348394.34839
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 2368, ( :1/2), 1)(2,\ 2368,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.63798473010.6379847301
L(12)L(\frac12) ≈\approx 0.63798473010.6379847301
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
37 1−T 1 - T
good3 1+pT2 1 + p T^{2}
5 1+2T+pT2 1 + 2 T + p T^{2}
7 1+4T+pT2 1 + 4 T + p T^{2}
11 1+pT2 1 + p T^{2}
13 1+2T+pT2 1 + 2 T + p T^{2}
17 1−2T+pT2 1 - 2 T + p T^{2}
19 1−2T+pT2 1 - 2 T + p T^{2}
23 1+6T+pT2 1 + 6 T + p T^{2}
29 1−6T+pT2 1 - 6 T + p T^{2}
31 1+2T+pT2 1 + 2 T + p T^{2}
41 1+2T+pT2 1 + 2 T + p T^{2}
43 1−2T+pT2 1 - 2 T + p T^{2}
47 1+4T+pT2 1 + 4 T + p T^{2}
53 1−6T+pT2 1 - 6 T + p T^{2}
59 1+6T+pT2 1 + 6 T + p T^{2}
61 1−6T+pT2 1 - 6 T + p T^{2}
67 1−8T+pT2 1 - 8 T + p T^{2}
71 1+4T+pT2 1 + 4 T + p T^{2}
73 1−14T+pT2 1 - 14 T + p T^{2}
79 1−2T+pT2 1 - 2 T + p T^{2}
83 1−4T+pT2 1 - 4 T + p T^{2}
89 1+6T+pT2 1 + 6 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

−8.973965603903636884920497152678, −8.139963688567899948562200386516, −7.53219516094291790010148313731, −6.59864521555049293791212644521, −5.97021818307546082272950549982, −5.06427273115980897713846035422, −3.87796534605450220035734187243, −3.30488835115774555617471587930, −2.42538163944277566108766365040, −0.48051744224849779476855570293, 0.48051744224849779476855570293, 2.42538163944277566108766365040, 3.30488835115774555617471587930, 3.87796534605450220035734187243, 5.06427273115980897713846035422, 5.97021818307546082272950549982, 6.59864521555049293791212644521, 7.53219516094291790010148313731, 8.139963688567899948562200386516, 8.973965603903636884920497152678

Graph of the ZZ-function along the critical line