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Properties

Label 2-5175-1.1-c1-0-31
Degree 22
Conductor 51755175
Sign 11
Analytic cond. 41.322541.3225
Root an. cond. 6.428266.42826
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  − 2.63·2-s + 4.96·4-s + 3.29·7-s − 7.82·8-s − 5.06·11-s − 1.39·13-s − 8.68·14-s + 10.7·16-s + 6.93·17-s − 3.60·19-s + 13.3·22-s + 23-s + 3.67·26-s + 16.3·28-s − 2.05·29-s − 3.78·31-s − 12.6·32-s − 18.3·34-s + 11.2·37-s + 9.51·38-s − 1.42·41-s − 7.57·43-s − 25.1·44-s − 2.63·46-s − 0.937·47-s + 3.83·49-s − 6.90·52-s + ⋯
L(s)  = 1  − 1.86·2-s + 2.48·4-s + 1.24·7-s − 2.76·8-s − 1.52·11-s − 0.385·13-s − 2.32·14-s + 2.68·16-s + 1.68·17-s − 0.826·19-s + 2.84·22-s + 0.208·23-s + 0.719·26-s + 3.08·28-s − 0.382·29-s − 0.680·31-s − 2.23·32-s − 3.14·34-s + 1.84·37-s + 1.54·38-s − 0.222·41-s − 1.15·43-s − 3.78·44-s − 0.389·46-s − 0.136·47-s + 0.547·49-s − 0.957·52-s + ⋯

Functional equation

Λ(s)=(5175s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 5175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(5175s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 5175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 51755175    =    32⋅52⋅233^{2} \cdot 5^{2} \cdot 23
Sign: 11
Analytic conductor: 41.322541.3225
Root analytic conductor: 6.428266.42826
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 5175, ( :1/2), 1)(2,\ 5175,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.75385578730.7538557873
L(12)L(\frac12) ≈\approx 0.75385578730.7538557873
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 1
5 1 1
23 1−T 1 - T
good2 1+2.63T+2T2 1 + 2.63T + 2T^{2}
7 1−3.29T+7T2 1 - 3.29T + 7T^{2}
11 1+5.06T+11T2 1 + 5.06T + 11T^{2}
13 1+1.39T+13T2 1 + 1.39T + 13T^{2}
17 1−6.93T+17T2 1 - 6.93T + 17T^{2}
19 1+3.60T+19T2 1 + 3.60T + 19T^{2}
29 1+2.05T+29T2 1 + 2.05T + 29T^{2}
31 1+3.78T+31T2 1 + 3.78T + 31T^{2}
37 1−11.2T+37T2 1 - 11.2T + 37T^{2}
41 1+1.42T+41T2 1 + 1.42T + 41T^{2}
43 1+7.57T+43T2 1 + 7.57T + 43T^{2}
47 1+0.937T+47T2 1 + 0.937T + 47T^{2}
53 1+0.143T+53T2 1 + 0.143T + 53T^{2}
59 1−3.20T+59T2 1 - 3.20T + 59T^{2}
61 1−5.22T+61T2 1 - 5.22T + 61T^{2}
67 1+14.7T+67T2 1 + 14.7T + 67T^{2}
71 1+3.57T+71T2 1 + 3.57T + 71T^{2}
73 1+0.0659T+73T2 1 + 0.0659T + 73T^{2}
79 1−16.3T+79T2 1 - 16.3T + 79T^{2}
83 1+0.439T+83T2 1 + 0.439T + 83T^{2}
89 1−8.21T+89T2 1 - 8.21T + 89T^{2}
97 1−9.42T+97T2 1 - 9.42T + 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

−8.112076770398107905898928161091, −7.72884973260377371141164882884, −7.35820406567662238869430474819, −6.24487722548974570215745593970, −5.47104534662245550967511810542, −4.74185588073650037970721180861, −3.28995321715952124922929882832, −2.38344957877817988169252468426, −1.69201810164252663486875508493, −0.62605450261852490887545447240, 0.62605450261852490887545447240, 1.69201810164252663486875508493, 2.38344957877817988169252468426, 3.28995321715952124922929882832, 4.74185588073650037970721180861, 5.47104534662245550967511810542, 6.24487722548974570215745593970, 7.35820406567662238869430474819, 7.72884973260377371141164882884, 8.112076770398107905898928161091

Graph of the ZZ-function along the critical line