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Properties

Label 2-912-1.1-c5-0-5
Degree 22
Conductor 912912
Sign 11
Analytic cond. 146.270146.270
Root an. cond. 12.094212.0942
Motivic weight 55
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 9·3-s − 54·5-s − 104·7-s + 81·9-s + 330·11-s − 46·13-s + 486·15-s − 618·17-s − 361·19-s + 936·21-s + 402·23-s − 209·25-s − 729·27-s − 2.62e3·29-s + 2.36e3·31-s − 2.97e3·33-s + 5.61e3·35-s − 1.21e4·37-s + 414·39-s − 1.88e4·41-s + 1.04e4·43-s − 4.37e3·45-s + 4.77e3·47-s − 5.99e3·49-s + 5.56e3·51-s − 1.94e4·53-s − 1.78e4·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.965·5-s − 0.802·7-s + 1/3·9-s + 0.822·11-s − 0.0754·13-s + 0.557·15-s − 0.518·17-s − 0.229·19-s + 0.463·21-s + 0.158·23-s − 0.0668·25-s − 0.192·27-s − 0.580·29-s + 0.442·31-s − 0.474·33-s + 0.774·35-s − 1.45·37-s + 0.0435·39-s − 1.75·41-s + 0.858·43-s − 0.321·45-s + 0.314·47-s − 0.356·49-s + 0.299·51-s − 0.951·53-s − 0.794·55-s + ⋯

Functional equation

Λ(s)=(912s/2 ΓC(s) L(s)=( Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(912s/2 ΓC(s+5/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 912912    =    24⋅3⋅192^{4} \cdot 3 \cdot 19
Sign: 11
Analytic conductor: 146.270146.270
Root analytic conductor: 12.094212.0942
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 912, ( :5/2), 1)(2,\ 912,\ (\ :5/2),\ 1)

Particular Values

L(3)L(3) ≈\approx 0.52792993150.5279299315
L(12)L(\frac12) ≈\approx 0.52792993150.5279299315
L(72)L(\frac{7}{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
3 1+p2T 1 + p^{2} T
19 1+p2T 1 + p^{2} T
good5 1+54T+p5T2 1 + 54 T + p^{5} T^{2}
7 1+104T+p5T2 1 + 104 T + p^{5} T^{2}
11 1−30pT+p5T2 1 - 30 p T + p^{5} T^{2}
13 1+46T+p5T2 1 + 46 T + p^{5} T^{2}
17 1+618T+p5T2 1 + 618 T + p^{5} T^{2}
23 1−402T+p5T2 1 - 402 T + p^{5} T^{2}
29 1+2628T+p5T2 1 + 2628 T + p^{5} T^{2}
31 1−2368T+p5T2 1 - 2368 T + p^{5} T^{2}
37 1+12130T+p5T2 1 + 12130 T + p^{5} T^{2}
41 1+18864T+p5T2 1 + 18864 T + p^{5} T^{2}
43 1−10408T+p5T2 1 - 10408 T + p^{5} T^{2}
47 1−4770T+p5T2 1 - 4770 T + p^{5} T^{2}
53 1+19452T+p5T2 1 + 19452 T + p^{5} T^{2}
59 1+30528T+p5T2 1 + 30528 T + p^{5} T^{2}
61 1−11138T+p5T2 1 - 11138 T + p^{5} T^{2}
67 1+49508T+p5T2 1 + 49508 T + p^{5} T^{2}
71 1+7572T+p5T2 1 + 7572 T + p^{5} T^{2}
73 1−2342T+p5T2 1 - 2342 T + p^{5} T^{2}
79 1+22424T+p5T2 1 + 22424 T + p^{5} T^{2}
83 1−46734T+p5T2 1 - 46734 T + p^{5} T^{2}
89 1+70104T+p5T2 1 + 70104 T + p^{5} T^{2}
97 1−105710T+p5T2 1 - 105710 T + p^{5} 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

−9.344609162207410367768075489974, −8.567015209724859307575485216278, −7.50771335869958083661693555558, −6.76230324248476894147518426848, −6.06362638032569195607839007589, −4.86992569219605215559856718215, −3.97697538226408777281271315033, −3.20294688703784200765036653309, −1.67268856680033577133971851547, −0.33116197167016676691995846755, 0.33116197167016676691995846755, 1.67268856680033577133971851547, 3.20294688703784200765036653309, 3.97697538226408777281271315033, 4.86992569219605215559856718215, 6.06362638032569195607839007589, 6.76230324248476894147518426848, 7.50771335869958083661693555558, 8.567015209724859307575485216278, 9.344609162207410367768075489974

Graph of the ZZ-function along the critical line