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Properties

Label 2-87e2-1.1-c1-0-43
Degree 22
Conductor 75697569
Sign 11
Analytic cond. 60.438760.4387
Root an. cond. 7.774237.77423
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  − 1.83·2-s + 1.36·4-s + 2.25·5-s − 0.744·7-s + 1.16·8-s − 4.14·10-s − 6.30·11-s − 1.64·13-s + 1.36·14-s − 4.86·16-s − 4.16·17-s + 1.04·19-s + 3.08·20-s + 11.5·22-s + 4.57·23-s + 0.100·25-s + 3.01·26-s − 1.01·28-s − 3.53·31-s + 6.60·32-s + 7.64·34-s − 1.68·35-s − 1.07·37-s − 1.90·38-s + 2.62·40-s + 1.20·41-s − 12.8·43-s + ⋯
L(s)  = 1  − 1.29·2-s + 0.683·4-s + 1.01·5-s − 0.281·7-s + 0.411·8-s − 1.31·10-s − 1.90·11-s − 0.456·13-s + 0.364·14-s − 1.21·16-s − 1.01·17-s + 0.238·19-s + 0.689·20-s + 2.46·22-s + 0.953·23-s + 0.0201·25-s + 0.592·26-s − 0.192·28-s − 0.634·31-s + 1.16·32-s + 1.31·34-s − 0.284·35-s − 0.176·37-s − 0.309·38-s + 0.415·40-s + 0.187·41-s − 1.95·43-s + ⋯

Functional equation

Λ(s)=(7569s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 7569 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(7569s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 7569 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 75697569    =    32⋅2923^{2} \cdot 29^{2}
Sign: 11
Analytic conductor: 60.438760.4387
Root analytic conductor: 7.774237.77423
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 7569, ( :1/2), 1)(2,\ 7569,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.56918726990.5691872699
L(12)L(\frac12) ≈\approx 0.56918726990.5691872699
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
29 1 1
good2 1+1.83T+2T2 1 + 1.83T + 2T^{2}
5 1−2.25T+5T2 1 - 2.25T + 5T^{2}
7 1+0.744T+7T2 1 + 0.744T + 7T^{2}
11 1+6.30T+11T2 1 + 6.30T + 11T^{2}
13 1+1.64T+13T2 1 + 1.64T + 13T^{2}
17 1+4.16T+17T2 1 + 4.16T + 17T^{2}
19 1−1.04T+19T2 1 - 1.04T + 19T^{2}
23 1−4.57T+23T2 1 - 4.57T + 23T^{2}
31 1+3.53T+31T2 1 + 3.53T + 31T^{2}
37 1+1.07T+37T2 1 + 1.07T + 37T^{2}
41 1−1.20T+41T2 1 - 1.20T + 41T^{2}
43 1+12.8T+43T2 1 + 12.8T + 43T^{2}
47 1−7.88T+47T2 1 - 7.88T + 47T^{2}
53 1−13.1T+53T2 1 - 13.1T + 53T^{2}
59 1+11.0T+59T2 1 + 11.0T + 59T^{2}
61 1+6.44T+61T2 1 + 6.44T + 61T^{2}
67 1+6.87T+67T2 1 + 6.87T + 67T^{2}
71 1−3.27T+71T2 1 - 3.27T + 71T^{2}
73 1−7.09T+73T2 1 - 7.09T + 73T^{2}
79 1−6.38T+79T2 1 - 6.38T + 79T^{2}
83 1−5.15T+83T2 1 - 5.15T + 83T^{2}
89 1+9.31T+89T2 1 + 9.31T + 89T^{2}
97 1+15.1T+97T2 1 + 15.1T + 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

−7.979826318912860520544191456228, −7.31666833259966518142164801982, −6.78112065584042566795983837303, −5.83264945669465112251807418350, −5.14798609332655718258099791678, −4.56217328703907305928760827145, −3.15851709053142453697335680255, −2.35446360535370328444529632957, −1.76814497460022526539607501279, −0.44373090913206291325624697759, 0.44373090913206291325624697759, 1.76814497460022526539607501279, 2.35446360535370328444529632957, 3.15851709053142453697335680255, 4.56217328703907305928760827145, 5.14798609332655718258099791678, 5.83264945669465112251807418350, 6.78112065584042566795983837303, 7.31666833259966518142164801982, 7.979826318912860520544191456228

Graph of the ZZ-function along the critical line