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Properties

Label 2-352-1.1-c1-0-0
Degree 22
Conductor 352352
Sign 11
Analytic cond. 2.810732.81073
Root an. cond. 1.676521.67652
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  − 2.56·3-s − 0.561·5-s + 3.56·9-s − 11-s + 2·13-s + 1.43·15-s + 7.12·17-s + 1.12·19-s + 7.68·23-s − 4.68·25-s − 1.43·27-s + 7.12·29-s + 5.43·31-s + 2.56·33-s − 5.68·37-s − 5.12·39-s − 8.24·41-s + 1.12·43-s − 2.00·45-s + 4·47-s − 7·49-s − 18.2·51-s + 8.24·53-s + 0.561·55-s − 2.87·57-s + 0.315·59-s + 9.36·61-s + ⋯
L(s)  = 1  − 1.47·3-s − 0.251·5-s + 1.18·9-s − 0.301·11-s + 0.554·13-s + 0.371·15-s + 1.72·17-s + 0.257·19-s + 1.60·23-s − 0.936·25-s − 0.276·27-s + 1.32·29-s + 0.976·31-s + 0.445·33-s − 0.934·37-s − 0.820·39-s − 1.28·41-s + 0.171·43-s − 0.298·45-s + 0.583·47-s − 49-s − 2.55·51-s + 1.13·53-s + 0.0757·55-s − 0.381·57-s + 0.0410·59-s + 1.19·61-s + ⋯

Functional equation

Λ(s)=(352s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(352s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 352352    =    25⋅112^{5} \cdot 11
Sign: 11
Analytic conductor: 2.810732.81073
Root analytic conductor: 1.676521.67652
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 352, ( :1/2), 1)(2,\ 352,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.80621434060.8062143406
L(12)L(\frac12) ≈\approx 0.80621434060.8062143406
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
11 1+T 1 + T
good3 1+2.56T+3T2 1 + 2.56T + 3T^{2}
5 1+0.561T+5T2 1 + 0.561T + 5T^{2}
7 1+7T2 1 + 7T^{2}
13 1−2T+13T2 1 - 2T + 13T^{2}
17 1−7.12T+17T2 1 - 7.12T + 17T^{2}
19 1−1.12T+19T2 1 - 1.12T + 19T^{2}
23 1−7.68T+23T2 1 - 7.68T + 23T^{2}
29 1−7.12T+29T2 1 - 7.12T + 29T^{2}
31 1−5.43T+31T2 1 - 5.43T + 31T^{2}
37 1+5.68T+37T2 1 + 5.68T + 37T^{2}
41 1+8.24T+41T2 1 + 8.24T + 41T^{2}
43 1−1.12T+43T2 1 - 1.12T + 43T^{2}
47 1−4T+47T2 1 - 4T + 47T^{2}
53 1−8.24T+53T2 1 - 8.24T + 53T^{2}
59 1−0.315T+59T2 1 - 0.315T + 59T^{2}
61 1−9.36T+61T2 1 - 9.36T + 61T^{2}
67 1+7.68T+67T2 1 + 7.68T + 67T^{2}
71 1+15.6T+71T2 1 + 15.6T + 71T^{2}
73 1+6T+73T2 1 + 6T + 73T^{2}
79 1−13.1T+79T2 1 - 13.1T + 79T^{2}
83 1−11.3T+83T2 1 - 11.3T + 83T^{2}
89 1−0.561T+89T2 1 - 0.561T + 89T^{2}
97 1−5.68T+97T2 1 - 5.68T + 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

−11.71637930315705832383908896038, −10.55797848962274206480390322282, −10.06589824870708370937970806103, −8.639658734738239079781631052209, −7.52032751894912668633406699865, −6.51405201310665444503786570704, −5.57052142861751043992859836001, −4.80128707043152207413273055438, −3.30198380786633657028182233539, −1.00310502704205625408320475655, 1.00310502704205625408320475655, 3.30198380786633657028182233539, 4.80128707043152207413273055438, 5.57052142861751043992859836001, 6.51405201310665444503786570704, 7.52032751894912668633406699865, 8.639658734738239079781631052209, 10.06589824870708370937970806103, 10.55797848962274206480390322282, 11.71637930315705832383908896038

Graph of the ZZ-function along the critical line