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Properties

Label 2-1152-8.5-c1-0-10
Degree 22
Conductor 11521152
Sign 0.707+0.707i0.707 + 0.707i
Analytic cond. 9.198769.19876
Root an. cond. 3.032943.03294
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·7-s + 4i·11-s − 4i·13-s + 2·17-s − 4i·19-s + 8·23-s + 5·25-s − 8i·29-s + 4·31-s − 4i·37-s + 6·41-s − 4i·43-s − 8·47-s + 9·49-s + 8i·53-s + ⋯
L(s)  = 1  − 1.51·7-s + 1.20i·11-s − 1.10i·13-s + 0.485·17-s − 0.917i·19-s + 1.66·23-s + 25-s − 1.48i·29-s + 0.718·31-s − 0.657i·37-s + 0.937·41-s − 0.609i·43-s − 1.16·47-s + 1.28·49-s + 1.09i·53-s + ⋯

Functional equation

Λ(s)=(1152s/2 ΓC(s) L(s)=((0.707+0.707i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1152s/2 ΓC(s+1/2) L(s)=((0.707+0.707i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, 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: 9.198769.19876
Root analytic conductor: 3.032943.03294
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1152(577,⋅)\chi_{1152} (577, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1152, ( :1/2), 0.707+0.707i)(2,\ 1152,\ (\ :1/2),\ 0.707 + 0.707i)

Particular Values

L(1)L(1) ≈\approx 1.2602617351.260261735
L(12)L(\frac12) ≈\approx 1.2602617351.260261735
L(32)L(\frac{3}{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 1−5T2 1 - 5T^{2}
7 1+4T+7T2 1 + 4T + 7T^{2}
11 1−4iT−11T2 1 - 4iT - 11T^{2}
13 1+4iT−13T2 1 + 4iT - 13T^{2}
17 1−2T+17T2 1 - 2T + 17T^{2}
19 1+4iT−19T2 1 + 4iT - 19T^{2}
23 1−8T+23T2 1 - 8T + 23T^{2}
29 1+8iT−29T2 1 + 8iT - 29T^{2}
31 1−4T+31T2 1 - 4T + 31T^{2}
37 1+4iT−37T2 1 + 4iT - 37T^{2}
41 1−6T+41T2 1 - 6T + 41T^{2}
43 1+4iT−43T2 1 + 4iT - 43T^{2}
47 1+8T+47T2 1 + 8T + 47T^{2}
53 1−8iT−53T2 1 - 8iT - 53T^{2}
59 1+12iT−59T2 1 + 12iT - 59T^{2}
61 1+12iT−61T2 1 + 12iT - 61T^{2}
67 1−12iT−67T2 1 - 12iT - 67T^{2}
71 1+8T+71T2 1 + 8T + 71T^{2}
73 1−6T+73T2 1 - 6T + 73T^{2}
79 1−4T+79T2 1 - 4T + 79T^{2}
83 1−4iT−83T2 1 - 4iT - 83T^{2}
89 1+6T+89T2 1 + 6T + 89T^{2}
97 1+2T+97T2 1 + 2T + 97T^{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

−9.663811347489229773588930741192, −9.132048518469446966080876486697, −7.973267919742689425604578360882, −7.08041437063497275499251406500, −6.50225767569213131207571378596, −5.44173536080275077641181238376, −4.52072352696192125743727723929, −3.27450664674083679853318110082, −2.57591982122599698652948702223, −0.66174165291503170838723340308, 1.12088616122945313007546739541, 2.96515159549670579511462281210, 3.43490745113121803273750169249, 4.73336276285199691977408898635, 5.85632511485663524756946117181, 6.55201621195319094357741740463, 7.21260709704559913010211181603, 8.502765385037461502213041733583, 9.062541882330379183500954998836, 9.836215025557699865033652175394

Graph of the ZZ-function along the critical line