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Properties

Label 2-87e2-1.1-c1-0-167
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  + 2.08·2-s + 2.34·4-s + 0.992·5-s + 2.45·7-s + 0.726·8-s + 2.06·10-s + 0.745·11-s + 0.730·13-s + 5.11·14-s − 3.18·16-s − 4.76·17-s + 1.38·19-s + 2.33·20-s + 1.55·22-s + 5.31·23-s − 4.01·25-s + 1.52·26-s + 5.75·28-s − 0.563·31-s − 8.08·32-s − 9.93·34-s + 2.43·35-s + 9.49·37-s + 2.88·38-s + 0.721·40-s + 5.63·41-s + 11.0·43-s + ⋯
L(s)  = 1  + 1.47·2-s + 1.17·4-s + 0.443·5-s + 0.926·7-s + 0.256·8-s + 0.654·10-s + 0.224·11-s + 0.202·13-s + 1.36·14-s − 0.795·16-s − 1.15·17-s + 0.317·19-s + 0.521·20-s + 0.331·22-s + 1.10·23-s − 0.803·25-s + 0.298·26-s + 1.08·28-s − 0.101·31-s − 1.42·32-s − 1.70·34-s + 0.411·35-s + 1.56·37-s + 0.467·38-s + 0.114·40-s + 0.880·41-s + 1.67·43-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 5.8903018695.890301869
L(12)L(\frac12) ≈\approx 5.8903018695.890301869
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 1−2.08T+2T2 1 - 2.08T + 2T^{2}
5 1−0.992T+5T2 1 - 0.992T + 5T^{2}
7 1−2.45T+7T2 1 - 2.45T + 7T^{2}
11 1−0.745T+11T2 1 - 0.745T + 11T^{2}
13 1−0.730T+13T2 1 - 0.730T + 13T^{2}
17 1+4.76T+17T2 1 + 4.76T + 17T^{2}
19 1−1.38T+19T2 1 - 1.38T + 19T^{2}
23 1−5.31T+23T2 1 - 5.31T + 23T^{2}
31 1+0.563T+31T2 1 + 0.563T + 31T^{2}
37 1−9.49T+37T2 1 - 9.49T + 37T^{2}
41 1−5.63T+41T2 1 - 5.63T + 41T^{2}
43 1−11.0T+43T2 1 - 11.0T + 43T^{2}
47 1−11.3T+47T2 1 - 11.3T + 47T^{2}
53 1−4.53T+53T2 1 - 4.53T + 53T^{2}
59 1−8.54T+59T2 1 - 8.54T + 59T^{2}
61 1−8.56T+61T2 1 - 8.56T + 61T^{2}
67 1−11.0T+67T2 1 - 11.0T + 67T^{2}
71 1+6.10T+71T2 1 + 6.10T + 71T^{2}
73 1−5.41T+73T2 1 - 5.41T + 73T^{2}
79 1+7.67T+79T2 1 + 7.67T + 79T^{2}
83 1+15.9T+83T2 1 + 15.9T + 83T^{2}
89 1+9.90T+89T2 1 + 9.90T + 89T^{2}
97 1−13.5T+97T2 1 - 13.5T + 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

−7.60134780594485029406262109264, −7.03248239979068281216089447343, −6.18849755153329320041791895990, −5.67578376742198882956753213847, −5.03918744605820972537825817952, −4.27275011037040106862817940095, −3.89936414732418839019875261911, −2.65278016016051110062197805639, −2.23701910420667289000917495422, −1.00693681199066280822759296783, 1.00693681199066280822759296783, 2.23701910420667289000917495422, 2.65278016016051110062197805639, 3.89936414732418839019875261911, 4.27275011037040106862817940095, 5.03918744605820972537825817952, 5.67578376742198882956753213847, 6.18849755153329320041791895990, 7.03248239979068281216089447343, 7.60134780594485029406262109264

Graph of the ZZ-function along the critical line