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Properties

Label 2-9075-1.1-c1-0-153
Degree 22
Conductor 90759075
Sign 11
Analytic cond. 72.464272.4642
Root an. cond. 8.512598.51259
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.53·2-s + 3-s + 0.369·4-s + 1.53·6-s + 0.290·7-s − 2.51·8-s + 9-s + 0.369·12-s + 6.97·13-s + 0.447·14-s − 4.60·16-s + 4.78·17-s + 1.53·18-s − 7.75·19-s + 0.290·21-s − 4·23-s − 2.51·24-s + 10.7·26-s + 27-s + 0.107·28-s + 7.41·29-s + 6.34·31-s − 2.06·32-s + 7.36·34-s + 0.369·36-s + 3.41·37-s − 11.9·38-s + ⋯
L(s)  = 1  + 1.08·2-s + 0.577·3-s + 0.184·4-s + 0.628·6-s + 0.109·7-s − 0.887·8-s + 0.333·9-s + 0.106·12-s + 1.93·13-s + 0.119·14-s − 1.15·16-s + 1.16·17-s + 0.362·18-s − 1.77·19-s + 0.0634·21-s − 0.834·23-s − 0.512·24-s + 2.10·26-s + 0.192·27-s + 0.0202·28-s + 1.37·29-s + 1.13·31-s − 0.364·32-s + 1.26·34-s + 0.0615·36-s + 0.562·37-s − 1.93·38-s + ⋯

Functional equation

Λ(s)=(9075s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(9075s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 90759075    =    3⋅52⋅1123 \cdot 5^{2} \cdot 11^{2}
Sign: 11
Analytic conductor: 72.464272.4642
Root analytic conductor: 8.512598.51259
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 9075, ( :1/2), 1)(2,\ 9075,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 4.5781980944.578198094
L(12)L(\frac12) ≈\approx 4.5781980944.578198094
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)
bad3 1−T 1 - T
5 1 1
11 1 1
good2 1−1.53T+2T2 1 - 1.53T + 2T^{2}
7 1−0.290T+7T2 1 - 0.290T + 7T^{2}
13 1−6.97T+13T2 1 - 6.97T + 13T^{2}
17 1−4.78T+17T2 1 - 4.78T + 17T^{2}
19 1+7.75T+19T2 1 + 7.75T + 19T^{2}
23 1+4T+23T2 1 + 4T + 23T^{2}
29 1−7.41T+29T2 1 - 7.41T + 29T^{2}
31 1−6.34T+31T2 1 - 6.34T + 31T^{2}
37 1−3.41T+37T2 1 - 3.41T + 37T^{2}
41 1−7.41T+41T2 1 - 7.41T + 41T^{2}
43 1−0.290T+43T2 1 - 0.290T + 43T^{2}
47 1+5.26T+47T2 1 + 5.26T + 47T^{2}
53 1+5.75T+53T2 1 + 5.75T + 53T^{2}
59 1−3.60T+59T2 1 - 3.60T + 59T^{2}
61 1−6.68T+61T2 1 - 6.68T + 61T^{2}
67 1+6.15T+67T2 1 + 6.15T + 67T^{2}
71 1+5.07T+71T2 1 + 5.07T + 71T^{2}
73 1+1.12T+73T2 1 + 1.12T + 73T^{2}
79 1−0.921T+79T2 1 - 0.921T + 79T^{2}
83 1−1.70T+83T2 1 - 1.70T + 83T^{2}
89 1−4.34T+89T2 1 - 4.34T + 89T^{2}
97 1+4.68T+97T2 1 + 4.68T + 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

−7.969668280437890692009094287635, −6.72176784957749855626729729711, −6.15729711952447827858920300835, −5.82211335118011540854835073926, −4.63578581259931773563150557616, −4.29159704524203784988612238903, −3.51000449289901905732841120458, −2.96850132261302279008297676182, −1.97671941444564389074796256942, −0.873204115582791448545432409592, 0.873204115582791448545432409592, 1.97671941444564389074796256942, 2.96850132261302279008297676182, 3.51000449289901905732841120458, 4.29159704524203784988612238903, 4.63578581259931773563150557616, 5.82211335118011540854835073926, 6.15729711952447827858920300835, 6.72176784957749855626729729711, 7.969668280437890692009094287635

Graph of the ZZ-function along the critical line