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Properties

Label 2-616-1.1-c3-0-41
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  + 7.15·3-s − 9.75·5-s − 7·7-s + 24.2·9-s + 11·11-s + 1.08·13-s − 69.8·15-s − 95.3·17-s − 77.1·19-s − 50.1·21-s + 114.·23-s − 29.8·25-s − 19.5·27-s + 280.·29-s − 233.·31-s + 78.7·33-s + 68.2·35-s − 294.·37-s + 7.76·39-s − 136.·41-s − 312.·43-s − 236.·45-s − 476.·47-s + 49·49-s − 682.·51-s + 283.·53-s − 107.·55-s + ⋯
L(s)  = 1  + 1.37·3-s − 0.872·5-s − 0.377·7-s + 0.898·9-s + 0.301·11-s + 0.0231·13-s − 1.20·15-s − 1.36·17-s − 0.931·19-s − 0.520·21-s + 1.03·23-s − 0.238·25-s − 0.139·27-s + 1.79·29-s − 1.35·31-s + 0.415·33-s + 0.329·35-s − 1.31·37-s + 0.0318·39-s − 0.520·41-s − 1.10·43-s − 0.784·45-s − 1.47·47-s + 0.142·49-s − 1.87·51-s + 0.733·53-s − 0.263·55-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 1−7.15T+27T2 1 - 7.15T + 27T^{2}
5 1+9.75T+125T2 1 + 9.75T + 125T^{2}
13 1−1.08T+2.19e3T2 1 - 1.08T + 2.19e3T^{2}
17 1+95.3T+4.91e3T2 1 + 95.3T + 4.91e3T^{2}
19 1+77.1T+6.85e3T2 1 + 77.1T + 6.85e3T^{2}
23 1−114.T+1.21e4T2 1 - 114.T + 1.21e4T^{2}
29 1−280.T+2.43e4T2 1 - 280.T + 2.43e4T^{2}
31 1+233.T+2.97e4T2 1 + 233.T + 2.97e4T^{2}
37 1+294.T+5.06e4T2 1 + 294.T + 5.06e4T^{2}
41 1+136.T+6.89e4T2 1 + 136.T + 6.89e4T^{2}
43 1+312.T+7.95e4T2 1 + 312.T + 7.95e4T^{2}
47 1+476.T+1.03e5T2 1 + 476.T + 1.03e5T^{2}
53 1−283.T+1.48e5T2 1 - 283.T + 1.48e5T^{2}
59 1+706.T+2.05e5T2 1 + 706.T + 2.05e5T^{2}
61 1+25.3T+2.26e5T2 1 + 25.3T + 2.26e5T^{2}
67 1−1.06e3T+3.00e5T2 1 - 1.06e3T + 3.00e5T^{2}
71 1+869.T+3.57e5T2 1 + 869.T + 3.57e5T^{2}
73 1−197.T+3.89e5T2 1 - 197.T + 3.89e5T^{2}
79 1−821.T+4.93e5T2 1 - 821.T + 4.93e5T^{2}
83 1+668.T+5.71e5T2 1 + 668.T + 5.71e5T^{2}
89 1−542.T+7.04e5T2 1 - 542.T + 7.04e5T^{2}
97 1+1.70e3T+9.12e5T2 1 + 1.70e3T + 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

−9.546001743932536888070791681230, −8.663904814661975379525515397315, −8.354039109763948754866146141171, −7.19902732407976313883589631369, −6.51826133733614055246560577275, −4.81850391319363622175355855493, −3.83994431796268828603972400349, −3.06594020999220997094359959928, −1.89862928191657351545986347557, 0, 1.89862928191657351545986347557, 3.06594020999220997094359959928, 3.83994431796268828603972400349, 4.81850391319363622175355855493, 6.51826133733614055246560577275, 7.19902732407976313883589631369, 8.354039109763948754866146141171, 8.663904814661975379525515397315, 9.546001743932536888070791681230

Graph of the ZZ-function along the critical line