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Properties

Label 2-87e2-1.1-c1-0-95
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  + 0.0831·2-s − 1.99·4-s − 1.93·5-s + 0.343·7-s − 0.332·8-s − 0.160·10-s + 2.42·11-s + 3.63·13-s + 0.0285·14-s + 3.95·16-s + 6.15·17-s − 1.35·19-s + 3.84·20-s + 0.201·22-s + 3.46·23-s − 1.27·25-s + 0.301·26-s − 0.685·28-s + 10.9·31-s + 0.993·32-s + 0.511·34-s − 0.663·35-s − 3.08·37-s − 0.112·38-s + 0.640·40-s − 5.28·41-s − 0.00844·43-s + ⋯
L(s)  = 1  + 0.0588·2-s − 0.996·4-s − 0.863·5-s + 0.129·7-s − 0.117·8-s − 0.0507·10-s + 0.730·11-s + 1.00·13-s + 0.00764·14-s + 0.989·16-s + 1.49·17-s − 0.309·19-s + 0.860·20-s + 0.0429·22-s + 0.722·23-s − 0.254·25-s + 0.0592·26-s − 0.129·28-s + 1.97·31-s + 0.175·32-s + 0.0877·34-s − 0.112·35-s − 0.507·37-s − 0.0182·38-s + 0.101·40-s − 0.825·41-s − 0.00128·43-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 1.5578596211.557859621
L(12)L(\frac12) ≈\approx 1.5578596211.557859621
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 1−0.0831T+2T2 1 - 0.0831T + 2T^{2}
5 1+1.93T+5T2 1 + 1.93T + 5T^{2}
7 1−0.343T+7T2 1 - 0.343T + 7T^{2}
11 1−2.42T+11T2 1 - 2.42T + 11T^{2}
13 1−3.63T+13T2 1 - 3.63T + 13T^{2}
17 1−6.15T+17T2 1 - 6.15T + 17T^{2}
19 1+1.35T+19T2 1 + 1.35T + 19T^{2}
23 1−3.46T+23T2 1 - 3.46T + 23T^{2}
31 1−10.9T+31T2 1 - 10.9T + 31T^{2}
37 1+3.08T+37T2 1 + 3.08T + 37T^{2}
41 1+5.28T+41T2 1 + 5.28T + 41T^{2}
43 1+0.00844T+43T2 1 + 0.00844T + 43T^{2}
47 1−5.79T+47T2 1 - 5.79T + 47T^{2}
53 1−4.14T+53T2 1 - 4.14T + 53T^{2}
59 1+5.76T+59T2 1 + 5.76T + 59T^{2}
61 1+6.23T+61T2 1 + 6.23T + 61T^{2}
67 1−11.0T+67T2 1 - 11.0T + 67T^{2}
71 1+12.8T+71T2 1 + 12.8T + 71T^{2}
73 1−0.459T+73T2 1 - 0.459T + 73T^{2}
79 1+12.4T+79T2 1 + 12.4T + 79T^{2}
83 1+2.87T+83T2 1 + 2.87T + 83T^{2}
89 1−12.2T+89T2 1 - 12.2T + 89T^{2}
97 1+2.72T+97T2 1 + 2.72T + 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

−7.975290220886155502592716458771, −7.37351196471227834058832651596, −6.39221520778612307925936373271, −5.78741643664657817163740311876, −4.89701630995171847474963232563, −4.29335639596900699488492095342, −3.58731462954538532817590145292, −3.09551287850145345631067777469, −1.45888059965780656837024871940, −0.69538489021728697424033390715, 0.69538489021728697424033390715, 1.45888059965780656837024871940, 3.09551287850145345631067777469, 3.58731462954538532817590145292, 4.29335639596900699488492095342, 4.89701630995171847474963232563, 5.78741643664657817163740311876, 6.39221520778612307925936373271, 7.37351196471227834058832651596, 7.975290220886155502592716458771

Graph of the ZZ-function along the critical line