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Properties

Label 2-616-1.1-c3-0-13
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  − 2·3-s + 12.1·5-s − 7·7-s − 23·9-s + 11·11-s − 16.3·13-s − 24.3·15-s + 30.4·17-s + 43.1·19-s + 14·21-s + 138.·23-s + 22.9·25-s + 100·27-s − 18.9·29-s + 20.1·31-s − 22·33-s − 85.1·35-s + 83.3·37-s + 32.6·39-s − 277.·41-s + 379.·43-s − 279.·45-s + 137.·47-s + 49·49-s − 60.9·51-s + 6.99·53-s + 133.·55-s + ⋯
L(s)  = 1  − 0.384·3-s + 1.08·5-s − 0.377·7-s − 0.851·9-s + 0.301·11-s − 0.348·13-s − 0.418·15-s + 0.435·17-s + 0.521·19-s + 0.145·21-s + 1.25·23-s + 0.183·25-s + 0.712·27-s − 0.121·29-s + 0.116·31-s − 0.116·33-s − 0.411·35-s + 0.370·37-s + 0.134·39-s − 1.05·41-s + 1.34·43-s − 0.926·45-s + 0.425·47-s + 0.142·49-s − 0.167·51-s + 0.0181·53-s + 0.328·55-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 1.9067024111.906702411
L(12)L(\frac12) ≈\approx 1.9067024111.906702411
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+2T+27T2 1 + 2T + 27T^{2}
5 1−12.1T+125T2 1 - 12.1T + 125T^{2}
13 1+16.3T+2.19e3T2 1 + 16.3T + 2.19e3T^{2}
17 1−30.4T+4.91e3T2 1 - 30.4T + 4.91e3T^{2}
19 1−43.1T+6.85e3T2 1 - 43.1T + 6.85e3T^{2}
23 1−138.T+1.21e4T2 1 - 138.T + 1.21e4T^{2}
29 1+18.9T+2.43e4T2 1 + 18.9T + 2.43e4T^{2}
31 1−20.1T+2.97e4T2 1 - 20.1T + 2.97e4T^{2}
37 1−83.3T+5.06e4T2 1 - 83.3T + 5.06e4T^{2}
41 1+277.T+6.89e4T2 1 + 277.T + 6.89e4T^{2}
43 1−379.T+7.95e4T2 1 - 379.T + 7.95e4T^{2}
47 1−137.T+1.03e5T2 1 - 137.T + 1.03e5T^{2}
53 1−6.99T+1.48e5T2 1 - 6.99T + 1.48e5T^{2}
59 1+160.T+2.05e5T2 1 + 160.T + 2.05e5T^{2}
61 1−499.T+2.26e5T2 1 - 499.T + 2.26e5T^{2}
67 1−1.03e3T+3.00e5T2 1 - 1.03e3T + 3.00e5T^{2}
71 1+319.T+3.57e5T2 1 + 319.T + 3.57e5T^{2}
73 1+699.T+3.89e5T2 1 + 699.T + 3.89e5T^{2}
79 1−181.T+4.93e5T2 1 - 181.T + 4.93e5T^{2}
83 1−1.41e3T+5.71e5T2 1 - 1.41e3T + 5.71e5T^{2}
89 1−640.T+7.04e5T2 1 - 640.T + 7.04e5T^{2}
97 1−617.T+9.12e5T2 1 - 617.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

−10.15121835027089500990482766180, −9.425837093179689911178496907344, −8.698556021186746690859352456922, −7.43996449460838653420156295329, −6.43619475930488780151446256329, −5.70388581418520990994110108970, −4.95020085092151565868102830871, −3.37347017909953614712346188513, −2.31323576181308357441243259491, −0.840144098044661097349100324063, 0.840144098044661097349100324063, 2.31323576181308357441243259491, 3.37347017909953614712346188513, 4.95020085092151565868102830871, 5.70388581418520990994110108970, 6.43619475930488780151446256329, 7.43996449460838653420156295329, 8.698556021186746690859352456922, 9.425837093179689911178496907344, 10.15121835027089500990482766180

Graph of the ZZ-function along the critical line