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Properties

Label 2-3168-1.1-c1-0-19
Degree 22
Conductor 31683168
Sign 11
Analytic cond. 25.296625.2966
Root an. cond. 5.029575.02957
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 + 2·7-s + 11-s − 2·13-s + 2·19-s − 25-s + 8·29-s + 4·31-s + 4·35-s − 6·37-s + 4·41-s + 6·43-s + 8·47-s − 3·49-s + 6·53-s + 2·55-s + 4·59-s − 6·61-s − 4·65-s + 4·67-s + 8·71-s − 10·73-s + 2·77-s − 14·79-s + 8·83-s − 2·89-s − 4·91-s + ⋯
L(s)  = 1  + 0.894·5-s + 0.755·7-s + 0.301·11-s − 0.554·13-s + 0.458·19-s − 1/5·25-s + 1.48·29-s + 0.718·31-s + 0.676·35-s − 0.986·37-s + 0.624·41-s + 0.914·43-s + 1.16·47-s − 3/7·49-s + 0.824·53-s + 0.269·55-s + 0.520·59-s − 0.768·61-s − 0.496·65-s + 0.488·67-s + 0.949·71-s − 1.17·73-s + 0.227·77-s − 1.57·79-s + 0.878·83-s − 0.211·89-s − 0.419·91-s + ⋯

Functional equation

Λ(s)=(3168s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(3168s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 31683168    =    25⋅32⋅112^{5} \cdot 3^{2} \cdot 11
Sign: 11
Analytic conductor: 25.296625.2966
Root analytic conductor: 5.029575.02957
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 3168, ( :1/2), 1)(2,\ 3168,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.5450614172.545061417
L(12)L(\frac12) ≈\approx 2.5450614172.545061417
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
11 1−T 1 - T
good5 1−2T+pT2 1 - 2 T + p T^{2}
7 1−2T+pT2 1 - 2 T + p T^{2}
13 1+2T+pT2 1 + 2 T + p T^{2}
17 1+pT2 1 + p T^{2}
19 1−2T+pT2 1 - 2 T + p T^{2}
23 1+pT2 1 + p T^{2}
29 1−8T+pT2 1 - 8 T + p T^{2}
31 1−4T+pT2 1 - 4 T + p T^{2}
37 1+6T+pT2 1 + 6 T + p T^{2}
41 1−4T+pT2 1 - 4 T + p T^{2}
43 1−6T+pT2 1 - 6 T + p T^{2}
47 1−8T+pT2 1 - 8 T + p T^{2}
53 1−6T+pT2 1 - 6 T + p T^{2}
59 1−4T+pT2 1 - 4 T + p T^{2}
61 1+6T+pT2 1 + 6 T + p T^{2}
67 1−4T+pT2 1 - 4 T + p T^{2}
71 1−8T+pT2 1 - 8 T + p T^{2}
73 1+10T+pT2 1 + 10 T + p T^{2}
79 1+14T+pT2 1 + 14 T + p T^{2}
83 1−8T+pT2 1 - 8 T + p T^{2}
89 1+2T+pT2 1 + 2 T + p T^{2}
97 1−14T+pT2 1 - 14 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

−8.741432993876465800467764479640, −7.937839645410758533791863487956, −7.19306300393180880258724467565, −6.36356837214073433395665681088, −5.61034403221069151944874816863, −4.90005158239302952603845530456, −4.11178662790885532582576400158, −2.86651983749511535818482386419, −2.04996257344413336974576420239, −1.01951695186635853569797311441, 1.01951695186635853569797311441, 2.04996257344413336974576420239, 2.86651983749511535818482386419, 4.11178662790885532582576400158, 4.90005158239302952603845530456, 5.61034403221069151944874816863, 6.36356837214073433395665681088, 7.19306300393180880258724467565, 7.937839645410758533791863487956, 8.741432993876465800467764479640

Graph of the ZZ-function along the critical line