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Properties

Label 2-616-1.1-c3-0-2
Degree 22
Conductor 616616
Sign 11
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 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 6.46·3-s − 3.00·5-s − 7·7-s + 14.7·9-s − 11·11-s − 29.1·13-s + 19.4·15-s + 31.7·17-s − 81.9·19-s + 45.2·21-s − 163.·23-s − 115.·25-s + 79.0·27-s − 93.6·29-s − 55.7·31-s + 71.0·33-s + 21.0·35-s + 344.·37-s + 188.·39-s + 36.0·41-s − 175.·43-s − 44.3·45-s − 364.·47-s + 49·49-s − 205.·51-s + 71.2·53-s + 33.0·55-s + ⋯
L(s)  = 1  − 1.24·3-s − 0.268·5-s − 0.377·7-s + 0.546·9-s − 0.301·11-s − 0.621·13-s + 0.334·15-s + 0.453·17-s − 0.988·19-s + 0.470·21-s − 1.48·23-s − 0.927·25-s + 0.563·27-s − 0.599·29-s − 0.322·31-s + 0.374·33-s + 0.101·35-s + 1.53·37-s + 0.773·39-s + 0.137·41-s − 0.622·43-s − 0.146·45-s − 1.13·47-s + 0.142·49-s − 0.564·51-s + 0.184·53-s + 0.0810·55-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: 11
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: 00
Selberg data: (2, 616, ( :3/2), 1)(2,\ 616,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 0.50303883640.5030388364
L(12)L(\frac12) ≈\approx 0.50303883640.5030388364
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+6.46T+27T2 1 + 6.46T + 27T^{2}
5 1+3.00T+125T2 1 + 3.00T + 125T^{2}
13 1+29.1T+2.19e3T2 1 + 29.1T + 2.19e3T^{2}
17 1−31.7T+4.91e3T2 1 - 31.7T + 4.91e3T^{2}
19 1+81.9T+6.85e3T2 1 + 81.9T + 6.85e3T^{2}
23 1+163.T+1.21e4T2 1 + 163.T + 1.21e4T^{2}
29 1+93.6T+2.43e4T2 1 + 93.6T + 2.43e4T^{2}
31 1+55.7T+2.97e4T2 1 + 55.7T + 2.97e4T^{2}
37 1−344.T+5.06e4T2 1 - 344.T + 5.06e4T^{2}
41 1−36.0T+6.89e4T2 1 - 36.0T + 6.89e4T^{2}
43 1+175.T+7.95e4T2 1 + 175.T + 7.95e4T^{2}
47 1+364.T+1.03e5T2 1 + 364.T + 1.03e5T^{2}
53 1−71.2T+1.48e5T2 1 - 71.2T + 1.48e5T^{2}
59 1−454.T+2.05e5T2 1 - 454.T + 2.05e5T^{2}
61 1−347.T+2.26e5T2 1 - 347.T + 2.26e5T^{2}
67 1−381.T+3.00e5T2 1 - 381.T + 3.00e5T^{2}
71 1+605.T+3.57e5T2 1 + 605.T + 3.57e5T^{2}
73 1−664.T+3.89e5T2 1 - 664.T + 3.89e5T^{2}
79 1−70.8T+4.93e5T2 1 - 70.8T + 4.93e5T^{2}
83 1−375.T+5.71e5T2 1 - 375.T + 5.71e5T^{2}
89 1−1.32e3T+7.04e5T2 1 - 1.32e3T + 7.04e5T^{2}
97 1+808.T+9.12e5T2 1 + 808.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

−10.24456048895978241834145162686, −9.686061423008184340908037654633, −8.340971355084247973682191981906, −7.49218565482856623943246694395, −6.39190131572731178795871625003, −5.77968830155796660041921987094, −4.78868412263933812378928304156, −3.73524586636528176218300335826, −2.18373343349134501432629898457, −0.42574170301776120103939541110, 0.42574170301776120103939541110, 2.18373343349134501432629898457, 3.73524586636528176218300335826, 4.78868412263933812378928304156, 5.77968830155796660041921987094, 6.39190131572731178795871625003, 7.49218565482856623943246694395, 8.340971355084247973682191981906, 9.686061423008184340908037654633, 10.24456048895978241834145162686

Graph of the ZZ-function along the critical line