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Properties

Label 2-1152-8.5-c3-0-14
Degree 22
Conductor 11521152
Sign 0.707−0.707i0.707 - 0.707i
Analytic cond. 67.970267.9702
Root an. cond. 8.244408.24440
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 8i·5-s − 12·7-s + 12i·11-s + 20i·13-s − 62·17-s − 108i·19-s − 72·23-s + 61·25-s + 128i·29-s + 204·31-s + 96i·35-s + 228i·37-s + 22·41-s − 204i·43-s − 600·47-s + ⋯
L(s)  = 1  − 0.715i·5-s − 0.647·7-s + 0.328i·11-s + 0.426i·13-s − 0.884·17-s − 1.30i·19-s − 0.652·23-s + 0.487·25-s + 0.819i·29-s + 1.18·31-s + 0.463i·35-s + 1.01i·37-s + 0.0838·41-s − 0.723i·43-s − 1.86·47-s + ⋯

Functional equation

Λ(s)=(1152s/2 ΓC(s) L(s)=((0.707−0.707i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(1152s/2 ΓC(s+3/2) L(s)=((0.707−0.707i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 11521152    =    27⋅322^{7} \cdot 3^{2}
Sign: 0.707−0.707i0.707 - 0.707i
Analytic conductor: 67.970267.9702
Root analytic conductor: 8.244408.24440
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ1152(577,⋅)\chi_{1152} (577, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1152, ( :3/2), 0.707−0.707i)(2,\ 1152,\ (\ :3/2),\ 0.707 - 0.707i)

Particular Values

L(2)L(2) ≈\approx 1.3089626641.308962664
L(12)L(\frac12) ≈\approx 1.3089626641.308962664
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
3 1 1
good5 1+8iT−125T2 1 + 8iT - 125T^{2}
7 1+12T+343T2 1 + 12T + 343T^{2}
11 1−12iT−1.33e3T2 1 - 12iT - 1.33e3T^{2}
13 1−20iT−2.19e3T2 1 - 20iT - 2.19e3T^{2}
17 1+62T+4.91e3T2 1 + 62T + 4.91e3T^{2}
19 1+108iT−6.85e3T2 1 + 108iT - 6.85e3T^{2}
23 1+72T+1.21e4T2 1 + 72T + 1.21e4T^{2}
29 1−128iT−2.43e4T2 1 - 128iT - 2.43e4T^{2}
31 1−204T+2.97e4T2 1 - 204T + 2.97e4T^{2}
37 1−228iT−5.06e4T2 1 - 228iT - 5.06e4T^{2}
41 1−22T+6.89e4T2 1 - 22T + 6.89e4T^{2}
43 1+204iT−7.95e4T2 1 + 204iT - 7.95e4T^{2}
47 1+600T+1.03e5T2 1 + 600T + 1.03e5T^{2}
53 1−256iT−1.48e5T2 1 - 256iT - 1.48e5T^{2}
59 1−828iT−2.05e5T2 1 - 828iT - 2.05e5T^{2}
61 1+84iT−2.26e5T2 1 + 84iT - 2.26e5T^{2}
67 1+348iT−3.00e5T2 1 + 348iT - 3.00e5T^{2}
71 1−456T+3.57e5T2 1 - 456T + 3.57e5T^{2}
73 1−822T+3.89e5T2 1 - 822T + 3.89e5T^{2}
79 1−1.35e3T+4.93e5T2 1 - 1.35e3T + 4.93e5T^{2}
83 1−108iT−5.71e5T2 1 - 108iT - 5.71e5T^{2}
89 1−938T+7.04e5T2 1 - 938T + 7.04e5T^{2}
97 1−1.27e3T+9.12e5T2 1 - 1.27e3T + 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

−9.372308067208420583702570582174, −8.857629011517218420482599704623, −7.995401160258820462855624880220, −6.82003486529962340353003755327, −6.39728605477763677591452819786, −5.01690214404273040329892142925, −4.52410848442112373346624290907, −3.28264206524264046864205554789, −2.16873219993995176586819503290, −0.840451307738406061285524141273, 0.40144566781329716112066346806, 2.02358474661958223288999036246, 3.08850997195347398361109585304, 3.86877645535031687897918520987, 5.06765661646327287209967904845, 6.31103761353169741421812367486, 6.49449998065306935175984411500, 7.78577175595520483713130097474, 8.343062224615406398777490456370, 9.537632404995594383477327956415

Graph of the ZZ-function along the critical line