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Properties

Label 2-9075-1.1-c1-0-291
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  + 2.67·2-s + 3-s + 5.15·4-s + 2.67·6-s + 2.80·7-s + 8.44·8-s + 9-s + 5.15·12-s − 5.11·13-s + 7.50·14-s + 12.2·16-s + 4.54·17-s + 2.67·18-s + 4.57·19-s + 2.80·21-s − 4·23-s + 8.44·24-s − 13.6·26-s + 27-s + 14.4·28-s + 2.38·29-s − 0.962·31-s + 15.9·32-s + 12.1·34-s + 5.15·36-s − 1.61·37-s + 12.2·38-s + ⋯
L(s)  = 1  + 1.89·2-s + 0.577·3-s + 2.57·4-s + 1.09·6-s + 1.06·7-s + 2.98·8-s + 0.333·9-s + 1.48·12-s − 1.41·13-s + 2.00·14-s + 3.06·16-s + 1.10·17-s + 0.630·18-s + 1.04·19-s + 0.612·21-s − 0.834·23-s + 1.72·24-s − 2.68·26-s + 0.192·27-s + 2.73·28-s + 0.443·29-s − 0.172·31-s + 2.81·32-s + 2.08·34-s + 0.859·36-s − 0.265·37-s + 1.98·38-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 11.0835906211.08359062
L(12)L(\frac12) ≈\approx 11.0835906211.08359062
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 1−2.67T+2T2 1 - 2.67T + 2T^{2}
7 1−2.80T+7T2 1 - 2.80T + 7T^{2}
13 1+5.11T+13T2 1 + 5.11T + 13T^{2}
17 1−4.54T+17T2 1 - 4.54T + 17T^{2}
19 1−4.57T+19T2 1 - 4.57T + 19T^{2}
23 1+4T+23T2 1 + 4T + 23T^{2}
29 1−2.38T+29T2 1 - 2.38T + 29T^{2}
31 1+0.962T+31T2 1 + 0.962T + 31T^{2}
37 1+1.61T+37T2 1 + 1.61T + 37T^{2}
41 1−2.38T+41T2 1 - 2.38T + 41T^{2}
43 1−2.80T+43T2 1 - 2.80T + 43T^{2}
47 1−4.31T+47T2 1 - 4.31T + 47T^{2}
53 1−6.57T+53T2 1 - 6.57T + 53T^{2}
59 1+13.2T+59T2 1 + 13.2T + 59T^{2}
61 1+7.92T+61T2 1 + 7.92T + 61T^{2}
67 1+10.7T+67T2 1 + 10.7T + 67T^{2}
71 1+7.35T+71T2 1 + 7.35T + 71T^{2}
73 1−6.41T+73T2 1 - 6.41T + 73T^{2}
79 1+1.35T+79T2 1 + 1.35T + 79T^{2}
83 1+0.806T+83T2 1 + 0.806T + 83T^{2}
89 1+2.96T+89T2 1 + 2.96T + 89T^{2}
97 1−9.92T+97T2 1 - 9.92T + 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

−7.61828139325699117336046659473, −7.09984165986404169033854273957, −6.04263812970950156136323919279, −5.50010325320873716703442156154, −4.79002671290631610833740203557, −4.41398244738947921079902727578, −3.48285827296203539637603309202, −2.86039017852520544331053773583, −2.12674544853775588197824821184, −1.32750749544657828394095531713, 1.32750749544657828394095531713, 2.12674544853775588197824821184, 2.86039017852520544331053773583, 3.48285827296203539637603309202, 4.41398244738947921079902727578, 4.79002671290631610833740203557, 5.50010325320873716703442156154, 6.04263812970950156136323919279, 7.09984165986404169033854273957, 7.61828139325699117336046659473

Graph of the ZZ-function along the critical line