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Properties

Label 2-40e2-1.1-c1-0-3
Degree 22
Conductor 16001600
Sign 11
Analytic cond. 12.776012.7760
Root an. cond. 3.574363.57436
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.23·3-s + 0.763·7-s + 7.47·9-s − 2.47·21-s − 5.70·23-s − 14.4·27-s + 6·29-s − 4.47·41-s − 11.2·43-s + 13.7·47-s − 6.41·49-s + 13.4·61-s + 5.70·63-s + 8.18·67-s + 18.4·69-s + 24.4·81-s + 17.7·83-s − 19.4·87-s + 6·89-s + 18·101-s + 20.1·103-s + 6.29·107-s − 13.4·109-s + ⋯
L(s)  = 1  − 1.86·3-s + 0.288·7-s + 2.49·9-s − 0.539·21-s − 1.19·23-s − 2.78·27-s + 1.11·29-s − 0.698·41-s − 1.71·43-s + 1.99·47-s − 0.916·49-s + 1.71·61-s + 0.719·63-s + 0.999·67-s + 2.22·69-s + 2.71·81-s + 1.94·83-s − 2.08·87-s + 0.635·89-s + 1.79·101-s + 1.98·103-s + 0.608·107-s − 1.28·109-s + ⋯

Functional equation

Λ(s)=(1600s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1600s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 16001600    =    26⋅522^{6} \cdot 5^{2}
Sign: 11
Analytic conductor: 12.776012.7760
Root analytic conductor: 3.574363.57436
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1600, ( :1/2), 1)(2,\ 1600,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.80533392450.8053339245
L(12)L(\frac12) ≈\approx 0.80533392450.8053339245
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
5 1 1
good3 1+3.23T+3T2 1 + 3.23T + 3T^{2}
7 1−0.763T+7T2 1 - 0.763T + 7T^{2}
11 1+11T2 1 + 11T^{2}
13 1+13T2 1 + 13T^{2}
17 1+17T2 1 + 17T^{2}
19 1+19T2 1 + 19T^{2}
23 1+5.70T+23T2 1 + 5.70T + 23T^{2}
29 1−6T+29T2 1 - 6T + 29T^{2}
31 1+31T2 1 + 31T^{2}
37 1+37T2 1 + 37T^{2}
41 1+4.47T+41T2 1 + 4.47T + 41T^{2}
43 1+11.2T+43T2 1 + 11.2T + 43T^{2}
47 1−13.7T+47T2 1 - 13.7T + 47T^{2}
53 1+53T2 1 + 53T^{2}
59 1+59T2 1 + 59T^{2}
61 1−13.4T+61T2 1 - 13.4T + 61T^{2}
67 1−8.18T+67T2 1 - 8.18T + 67T^{2}
71 1+71T2 1 + 71T^{2}
73 1+73T2 1 + 73T^{2}
79 1+79T2 1 + 79T^{2}
83 1−17.7T+83T2 1 - 17.7T + 83T^{2}
89 1−6T+89T2 1 - 6T + 89T^{2}
97 1+97T2 1 + 97T^{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

−9.808634071172554708725211779728, −8.553190499172138405910183511041, −7.62561266042530916678711360244, −6.72016697862793868423748699054, −6.17364264952357770932618056764, −5.29369115774265646522695955691, −4.69630400834468146692411967977, −3.72895650149875186242656031563, −1.96314886680260062258064664354, −0.69687543080154082339848196756, 0.69687543080154082339848196756, 1.96314886680260062258064664354, 3.72895650149875186242656031563, 4.69630400834468146692411967977, 5.29369115774265646522695955691, 6.17364264952357770932618056764, 6.72016697862793868423748699054, 7.62561266042530916678711360244, 8.553190499172138405910183511041, 9.808634071172554708725211779728

Graph of the ZZ-function along the critical line