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Properties

Label 2-616-1.1-c3-0-37
Degree 22
Conductor 616616
Sign −1-1
Analytic cond. 36.345136.3451
Root an. cond. 6.028696.02869
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.18·3-s + 2.34·5-s + 7·7-s − 22.2·9-s − 11·11-s − 21.0·13-s + 5.12·15-s + 86.0·17-s − 129.·19-s + 15.3·21-s + 109.·23-s − 119.·25-s − 107.·27-s − 149.·29-s − 284.·31-s − 24.0·33-s + 16.4·35-s − 107.·37-s − 46.0·39-s + 505.·41-s + 26.8·43-s − 52.0·45-s − 12.1·47-s + 49·49-s + 188.·51-s − 571.·53-s − 25.7·55-s + ⋯
L(s)  = 1  + 0.421·3-s + 0.209·5-s + 0.377·7-s − 0.822·9-s − 0.301·11-s − 0.449·13-s + 0.0882·15-s + 1.22·17-s − 1.56·19-s + 0.159·21-s + 0.992·23-s − 0.956·25-s − 0.767·27-s − 0.957·29-s − 1.64·31-s − 0.126·33-s + 0.0792·35-s − 0.476·37-s − 0.189·39-s + 1.92·41-s + 0.0951·43-s − 0.172·45-s − 0.0377·47-s + 0.142·49-s + 0.517·51-s − 1.48·53-s − 0.0631·55-s + ⋯

Functional equation

Λ(s)=(616s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(616s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 616 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 616616    =    23⋅7⋅112^{3} \cdot 7 \cdot 11
Sign: −1-1
Analytic conductor: 36.345136.3451
Root analytic conductor: 6.028696.02869
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 616, ( :3/2), −1)(2,\ 616,\ (\ :3/2),\ -1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
L(52)L(\frac{5}{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
7 1−7T 1 - 7T
11 1+11T 1 + 11T
good3 1−2.18T+27T2 1 - 2.18T + 27T^{2}
5 1−2.34T+125T2 1 - 2.34T + 125T^{2}
13 1+21.0T+2.19e3T2 1 + 21.0T + 2.19e3T^{2}
17 1−86.0T+4.91e3T2 1 - 86.0T + 4.91e3T^{2}
19 1+129.T+6.85e3T2 1 + 129.T + 6.85e3T^{2}
23 1−109.T+1.21e4T2 1 - 109.T + 1.21e4T^{2}
29 1+149.T+2.43e4T2 1 + 149.T + 2.43e4T^{2}
31 1+284.T+2.97e4T2 1 + 284.T + 2.97e4T^{2}
37 1+107.T+5.06e4T2 1 + 107.T + 5.06e4T^{2}
41 1−505.T+6.89e4T2 1 - 505.T + 6.89e4T^{2}
43 1−26.8T+7.95e4T2 1 - 26.8T + 7.95e4T^{2}
47 1+12.1T+1.03e5T2 1 + 12.1T + 1.03e5T^{2}
53 1+571.T+1.48e5T2 1 + 571.T + 1.48e5T^{2}
59 1+53.0T+2.05e5T2 1 + 53.0T + 2.05e5T^{2}
61 1+469.T+2.26e5T2 1 + 469.T + 2.26e5T^{2}
67 1−732.T+3.00e5T2 1 - 732.T + 3.00e5T^{2}
71 1−338.T+3.57e5T2 1 - 338.T + 3.57e5T^{2}
73 1+1.14e3T+3.89e5T2 1 + 1.14e3T + 3.89e5T^{2}
79 1+156.T+4.93e5T2 1 + 156.T + 4.93e5T^{2}
83 1+960.T+5.71e5T2 1 + 960.T + 5.71e5T^{2}
89 1+466.T+7.04e5T2 1 + 466.T + 7.04e5T^{2}
97 1+126.T+9.12e5T2 1 + 126.T + 9.12e5T^{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

−9.640604134880118586568732573137, −8.946982887769362984662835010103, −8.014520242179649609807461754021, −7.34252774801274780904707278729, −5.98356511243440000664740212127, −5.29250387270163217201251430345, −4.00504752181082623528343567728, −2.86312927513021288933869135879, −1.79332899796971248873645869926, 0, 1.79332899796971248873645869926, 2.86312927513021288933869135879, 4.00504752181082623528343567728, 5.29250387270163217201251430345, 5.98356511243440000664740212127, 7.34252774801274780904707278729, 8.014520242179649609807461754021, 8.946982887769362984662835010103, 9.640604134880118586568732573137

Graph of the ZZ-function along the critical line