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Properties

Label 2-87e2-1.1-c1-0-105
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  − 0.183·2-s − 1.96·4-s + 3.26·5-s − 0.215·7-s + 0.726·8-s − 0.597·10-s − 2.92·11-s − 3.58·13-s + 0.0394·14-s + 3.79·16-s + 7.11·17-s + 1.38·19-s − 6.41·20-s + 0.535·22-s + 6.71·23-s + 5.63·25-s + 0.656·26-s + 0.423·28-s + 6.41·31-s − 2.14·32-s − 1.30·34-s − 0.702·35-s − 6.11·37-s − 0.253·38-s + 2.36·40-s + 6.50·41-s − 6.24·43-s + ⋯
L(s)  = 1  − 0.129·2-s − 0.983·4-s + 1.45·5-s − 0.0813·7-s + 0.256·8-s − 0.188·10-s − 0.881·11-s − 0.994·13-s + 0.0105·14-s + 0.949·16-s + 1.72·17-s + 0.317·19-s − 1.43·20-s + 0.114·22-s + 1.40·23-s + 1.12·25-s + 0.128·26-s + 0.0800·28-s + 1.15·31-s − 0.379·32-s − 0.223·34-s − 0.118·35-s − 1.00·37-s − 0.0410·38-s + 0.374·40-s + 1.01·41-s − 0.952·43-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 1.9173792131.917379213
L(12)L(\frac12) ≈\approx 1.9173792131.917379213
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+0.183T+2T2 1 + 0.183T + 2T^{2}
5 1−3.26T+5T2 1 - 3.26T + 5T^{2}
7 1+0.215T+7T2 1 + 0.215T + 7T^{2}
11 1+2.92T+11T2 1 + 2.92T + 11T^{2}
13 1+3.58T+13T2 1 + 3.58T + 13T^{2}
17 1−7.11T+17T2 1 - 7.11T + 17T^{2}
19 1−1.38T+19T2 1 - 1.38T + 19T^{2}
23 1−6.71T+23T2 1 - 6.71T + 23T^{2}
31 1−6.41T+31T2 1 - 6.41T + 31T^{2}
37 1+6.11T+37T2 1 + 6.11T + 37T^{2}
41 1−6.50T+41T2 1 - 6.50T + 41T^{2}
43 1+6.24T+43T2 1 + 6.24T + 43T^{2}
47 1−9.04T+47T2 1 - 9.04T + 47T^{2}
53 1+4.53T+53T2 1 + 4.53T + 53T^{2}
59 1+12.0T+59T2 1 + 12.0T + 59T^{2}
61 1−1.58T+61T2 1 - 1.58T + 61T^{2}
67 1+7.18T+67T2 1 + 7.18T + 67T^{2}
71 1−13.4T+71T2 1 - 13.4T + 71T^{2}
73 1+13.8T+73T2 1 + 13.8T + 73T^{2}
79 1+3.35T+79T2 1 + 3.35T + 79T^{2}
83 1−5.23T+83T2 1 - 5.23T + 83T^{2}
89 1−12.9T+89T2 1 - 12.9T + 89T^{2}
97 1−3.92T+97T2 1 - 3.92T + 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

−7.80834662326833449050351636504, −7.39498508216891641195528104096, −6.32815192246337735143905201295, −5.62016124008237595646927483670, −5.10015249548206518087687174302, −4.67541631078150860543762943081, −3.30658448986381622381857510957, −2.78854821328600252291656389107, −1.67368061418384315615505288318, −0.74042989821156927320148775086, 0.74042989821156927320148775086, 1.67368061418384315615505288318, 2.78854821328600252291656389107, 3.30658448986381622381857510957, 4.67541631078150860543762943081, 5.10015249548206518087687174302, 5.62016124008237595646927483670, 6.32815192246337735143905201295, 7.39498508216891641195528104096, 7.80834662326833449050351636504

Graph of the ZZ-function along the critical line