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Properties

Label 2-85e2-1.1-c1-0-116
Degree 22
Conductor 72257225
Sign 11
Analytic cond. 57.691957.6919
Root an. cond. 7.595517.59551
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-s − 1.41·3-s − 4-s − 1.41·6-s + 4.24·7-s − 3·8-s − 0.999·9-s + 4.24·11-s + 1.41·12-s + 4.24·14-s − 16-s − 0.999·18-s − 6·19-s − 6·21-s + 4.24·22-s − 1.41·23-s + 4.24·24-s + 5.65·27-s − 4.24·28-s + 4.24·29-s + 1.41·31-s + 5·32-s − 6·33-s + 0.999·36-s − 4.24·37-s − 6·38-s − 4.24·41-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.816·3-s − 0.5·4-s − 0.577·6-s + 1.60·7-s − 1.06·8-s − 0.333·9-s + 1.27·11-s + 0.408·12-s + 1.13·14-s − 0.250·16-s − 0.235·18-s − 1.37·19-s − 1.30·21-s + 0.904·22-s − 0.294·23-s + 0.866·24-s + 1.08·27-s − 0.801·28-s + 0.787·29-s + 0.254·31-s + 0.883·32-s − 1.04·33-s + 0.166·36-s − 0.697·37-s − 0.973·38-s − 0.662·41-s + ⋯

Functional equation

Λ(s)=(7225s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(7225s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 72257225    =    52⋅1725^{2} \cdot 17^{2}
Sign: 11
Analytic conductor: 57.691957.6919
Root analytic conductor: 7.595517.59551
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 7225, ( :1/2), 1)(2,\ 7225,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.9507146951.950714695
L(12)L(\frac12) ≈\approx 1.9507146951.950714695
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)
bad5 1 1
17 1 1
good2 1−T+2T2 1 - T + 2T^{2}
3 1+1.41T+3T2 1 + 1.41T + 3T^{2}
7 1−4.24T+7T2 1 - 4.24T + 7T^{2}
11 1−4.24T+11T2 1 - 4.24T + 11T^{2}
13 1+13T2 1 + 13T^{2}
19 1+6T+19T2 1 + 6T + 19T^{2}
23 1+1.41T+23T2 1 + 1.41T + 23T^{2}
29 1−4.24T+29T2 1 - 4.24T + 29T^{2}
31 1−1.41T+31T2 1 - 1.41T + 31T^{2}
37 1+4.24T+37T2 1 + 4.24T + 37T^{2}
41 1+4.24T+41T2 1 + 4.24T + 41T^{2}
43 1−12T+43T2 1 - 12T + 43T^{2}
47 1−2T+47T2 1 - 2T + 47T^{2}
53 1−2T+53T2 1 - 2T + 53T^{2}
59 1−6T+59T2 1 - 6T + 59T^{2}
61 1−1.41T+61T2 1 - 1.41T + 61T^{2}
67 1+6T+67T2 1 + 6T + 67T^{2}
71 1+4.24T+71T2 1 + 4.24T + 71T^{2}
73 1+4.24T+73T2 1 + 4.24T + 73T^{2}
79 1+9.89T+79T2 1 + 9.89T + 79T^{2}
83 1+4T+83T2 1 + 4T + 83T^{2}
89 1−6T+89T2 1 - 6T + 89T^{2}
97 1−4.24T+97T2 1 - 4.24T + 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

−8.034644587788928071412803740751, −6.99366954175650791454059751724, −6.25976830874351044227757554532, −5.71784662991099448350004152575, −5.04440987493722448484406740064, −4.36258996367297469945829769524, −4.03633826295134161834150902256, −2.80616085018079810668167867651, −1.72379836756367044463261245269, −0.68700246874957017145831642276, 0.68700246874957017145831642276, 1.72379836756367044463261245269, 2.80616085018079810668167867651, 4.03633826295134161834150902256, 4.36258996367297469945829769524, 5.04440987493722448484406740064, 5.71784662991099448350004152575, 6.25976830874351044227757554532, 6.99366954175650791454059751724, 8.034644587788928071412803740751

Graph of the ZZ-function along the critical line