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Properties

Label 2-352-1.1-c1-0-2
Degree 22
Conductor 352352
Sign 11
Analytic cond. 2.810732.81073
Root an. cond. 1.676521.67652
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  + 3-s + 5-s + 4·7-s − 2·9-s + 11-s − 2·13-s + 15-s − 2·19-s + 4·21-s + 9·23-s − 4·25-s − 5·27-s + 4·29-s + 5·31-s + 33-s + 4·35-s − 9·37-s − 2·39-s + 2·41-s − 6·43-s − 2·45-s − 4·47-s + 9·49-s − 6·53-s + 55-s − 2·57-s − 5·59-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 1.51·7-s − 2/3·9-s + 0.301·11-s − 0.554·13-s + 0.258·15-s − 0.458·19-s + 0.872·21-s + 1.87·23-s − 4/5·25-s − 0.962·27-s + 0.742·29-s + 0.898·31-s + 0.174·33-s + 0.676·35-s − 1.47·37-s − 0.320·39-s + 0.312·41-s − 0.914·43-s − 0.298·45-s − 0.583·47-s + 9/7·49-s − 0.824·53-s + 0.134·55-s − 0.264·57-s − 0.650·59-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: yes
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 1.8394225491.839422549
L(12)L(\frac12) ≈\approx 1.8394225491.839422549
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−T+pT2 1 - T + p T^{2}
5 1−T+pT2 1 - T + p T^{2}
7 1−4T+pT2 1 - 4 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 1−9T+pT2 1 - 9 T + p T^{2}
29 1−4T+pT2 1 - 4 T + p T^{2}
31 1−5T+pT2 1 - 5 T + p T^{2}
37 1+9T+pT2 1 + 9 T + p T^{2}
41 1−2T+pT2 1 - 2 T + p T^{2}
43 1+6T+pT2 1 + 6 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+5T+pT2 1 + 5 T + p T^{2}
61 1+pT2 1 + p T^{2}
67 1+13T+pT2 1 + 13 T + p T^{2}
71 1+T+pT2 1 + T + p T^{2}
73 1−14T+pT2 1 - 14 T + p T^{2}
79 1+10T+pT2 1 + 10 T + p T^{2}
83 1−14T+pT2 1 - 14 T + p T^{2}
89 1+13T+pT2 1 + 13 T + p T^{2}
97 1+19T+pT2 1 + 19 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

−11.43031674585287936480172767837, −10.65536531680425134672181778028, −9.477460186491048050604925394554, −8.595119251819881391518110548197, −7.956649639760417596283329475573, −6.77843662469483305850122200054, −5.43215102206894998656777956795, −4.57725485407179341808498131930, −2.97331376469435176368062512619, −1.71677882912616726095004967526, 1.71677882912616726095004967526, 2.97331376469435176368062512619, 4.57725485407179341808498131930, 5.43215102206894998656777956795, 6.77843662469483305850122200054, 7.956649639760417596283329475573, 8.595119251819881391518110548197, 9.477460186491048050604925394554, 10.65536531680425134672181778028, 11.43031674585287936480172767837

Graph of the ZZ-function along the critical line