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Properties

Label 2-57e2-1.1-c1-0-57
Degree 22
Conductor 32493249
Sign 11
Analytic cond. 25.943325.9433
Root an. cond. 5.093465.09346
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  − 2·4-s + 5-s + 3·7-s + 5·11-s + 4·16-s + 7·17-s − 2·20-s + 4·23-s − 4·25-s − 6·28-s + 3·35-s − 43-s − 10·44-s − 13·47-s + 2·49-s + 5·55-s + 15·61-s − 8·64-s − 14·68-s − 11·73-s + 15·77-s + 4·80-s + 16·83-s + 7·85-s − 8·92-s + 8·100-s − 10·101-s + ⋯
L(s)  = 1  − 4-s + 0.447·5-s + 1.13·7-s + 1.50·11-s + 16-s + 1.69·17-s − 0.447·20-s + 0.834·23-s − 4/5·25-s − 1.13·28-s + 0.507·35-s − 0.152·43-s − 1.50·44-s − 1.89·47-s + 2/7·49-s + 0.674·55-s + 1.92·61-s − 64-s − 1.69·68-s − 1.28·73-s + 1.70·77-s + 0.447·80-s + 1.75·83-s + 0.759·85-s − 0.834·92-s + 4/5·100-s − 0.995·101-s + ⋯

Functional equation

Λ(s)=(3249s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(3249s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 32493249    =    32⋅1923^{2} \cdot 19^{2}
Sign: 11
Analytic conductor: 25.943325.9433
Root analytic conductor: 5.093465.09346
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 3249, ( :1/2), 1)(2,\ 3249,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.2202076382.220207638
L(12)L(\frac12) ≈\approx 2.2202076382.220207638
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)
bad3 1 1
19 1 1
good2 1+pT2 1 + p T^{2}
5 1−T+pT2 1 - T + p T^{2}
7 1−3T+pT2 1 - 3 T + p T^{2}
11 1−5T+pT2 1 - 5 T + p T^{2}
13 1+pT2 1 + p T^{2}
17 1−7T+pT2 1 - 7 T + p T^{2}
23 1−4T+pT2 1 - 4 T + p T^{2}
29 1+pT2 1 + p T^{2}
31 1+pT2 1 + p T^{2}
37 1+pT2 1 + p T^{2}
41 1+pT2 1 + p T^{2}
43 1+T+pT2 1 + T + p T^{2}
47 1+13T+pT2 1 + 13 T + p T^{2}
53 1+pT2 1 + p T^{2}
59 1+pT2 1 + p T^{2}
61 1−15T+pT2 1 - 15 T + p T^{2}
67 1+pT2 1 + p T^{2}
71 1+pT2 1 + p T^{2}
73 1+11T+pT2 1 + 11 T + p T^{2}
79 1+pT2 1 + p T^{2}
83 1−16T+pT2 1 - 16 T + p T^{2}
89 1+pT2 1 + p T^{2}
97 1+pT2 1 + 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

−8.580042248187093986713935126037, −8.107116855348831176369596581034, −7.27943314894574780769453240025, −6.27245985312872964632310218742, −5.45289188915211295785069372208, −4.88187668285910676494600805335, −4.00355565153622805697617293535, −3.28074247116569086926038001604, −1.71787306481732958102463482448, −1.01885130548047428295164993318, 1.01885130548047428295164993318, 1.71787306481732958102463482448, 3.28074247116569086926038001604, 4.00355565153622805697617293535, 4.88187668285910676494600805335, 5.45289188915211295785069372208, 6.27245985312872964632310218742, 7.27943314894574780769453240025, 8.107116855348831176369596581034, 8.580042248187093986713935126037

Graph of the ZZ-function along the critical line