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Properties

Label 2-2736-1.1-c1-0-17
Degree 22
Conductor 27362736
Sign 11
Analytic cond. 21.847021.8470
Root an. cond. 4.674084.67408
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 + 6·13-s + 6·17-s + 19-s + 4·23-s − 25-s − 2·29-s − 8·31-s − 10·37-s + 2·41-s + 4·43-s + 12·47-s − 7·49-s + 6·53-s − 12·59-s − 2·61-s + 12·65-s + 4·67-s + 10·73-s + 16·83-s + 12·85-s + 2·89-s + 2·95-s + 10·97-s + 10·101-s − 8·103-s + 4·107-s + ⋯
L(s)  = 1  + 0.894·5-s + 1.66·13-s + 1.45·17-s + 0.229·19-s + 0.834·23-s − 1/5·25-s − 0.371·29-s − 1.43·31-s − 1.64·37-s + 0.312·41-s + 0.609·43-s + 1.75·47-s − 49-s + 0.824·53-s − 1.56·59-s − 0.256·61-s + 1.48·65-s + 0.488·67-s + 1.17·73-s + 1.75·83-s + 1.30·85-s + 0.211·89-s + 0.205·95-s + 1.01·97-s + 0.995·101-s − 0.788·103-s + 0.386·107-s + ⋯

Functional equation

Λ(s)=(2736s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(2736s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 27362736    =    24⋅32⋅192^{4} \cdot 3^{2} \cdot 19
Sign: 11
Analytic conductor: 21.847021.8470
Root analytic conductor: 4.674084.67408
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 2736, ( :1/2), 1)(2,\ 2736,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.5063982362.506398236
L(12)L(\frac12) ≈\approx 2.5063982362.506398236
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
19 1−T 1 - T
good5 1−2T+pT2 1 - 2 T + p T^{2}
7 1+pT2 1 + p T^{2}
11 1+pT2 1 + p T^{2}
13 1−6T+pT2 1 - 6 T + p T^{2}
17 1−6T+pT2 1 - 6 T + p T^{2}
23 1−4T+pT2 1 - 4 T + p T^{2}
29 1+2T+pT2 1 + 2 T + p T^{2}
31 1+8T+pT2 1 + 8 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 1−4T+pT2 1 - 4 T + p T^{2}
47 1−12T+pT2 1 - 12 T + p T^{2}
53 1−6T+pT2 1 - 6 T + p T^{2}
59 1+12T+pT2 1 + 12 T + p T^{2}
61 1+2T+pT2 1 + 2 T + p T^{2}
67 1−4T+pT2 1 - 4 T + p T^{2}
71 1+pT2 1 + p T^{2}
73 1−10T+pT2 1 - 10 T + p T^{2}
79 1+pT2 1 + p T^{2}
83 1−16T+pT2 1 - 16 T + p T^{2}
89 1−2T+pT2 1 - 2 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

−9.092693734190788229037074202448, −8.062008804214737453511932083588, −7.33667434445098454643296446245, −6.39584814801769229586264060019, −5.69178884171323828176864961556, −5.22523151161492305197654101068, −3.84859941092912106278451327058, −3.25148783575260438600086430315, −1.95255670636420425042320954209, −1.07628396925620245570845051253, 1.07628396925620245570845051253, 1.95255670636420425042320954209, 3.25148783575260438600086430315, 3.84859941092912106278451327058, 5.22523151161492305197654101068, 5.69178884171323828176864961556, 6.39584814801769229586264060019, 7.33667434445098454643296446245, 8.062008804214737453511932083588, 9.092693734190788229037074202448

Graph of the ZZ-function along the critical line