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Properties

Label 2-85e2-1.1-c1-0-393
Degree 22
Conductor 72257225
Sign −1-1
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 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.88·2-s + 1.88·3-s + 1.55·4-s + 3.56·6-s + 0.335·7-s − 0.836·8-s + 0.569·9-s − 3.80·11-s + 2.94·12-s + 0.558·13-s + 0.632·14-s − 4.69·16-s + 1.07·18-s − 3.73·19-s + 0.633·21-s − 7.18·22-s + 1.24·23-s − 1.58·24-s + 1.05·26-s − 4.59·27-s + 0.521·28-s − 3.98·29-s − 9.36·31-s − 7.17·32-s − 7.19·33-s + 0.887·36-s + 6.77·37-s + ⋯
L(s)  = 1  + 1.33·2-s + 1.09·3-s + 0.778·4-s + 1.45·6-s + 0.126·7-s − 0.295·8-s + 0.189·9-s − 1.14·11-s + 0.848·12-s + 0.154·13-s + 0.169·14-s − 1.17·16-s + 0.253·18-s − 0.856·19-s + 0.138·21-s − 1.53·22-s + 0.259·23-s − 0.322·24-s + 0.206·26-s − 0.883·27-s + 0.0986·28-s − 0.740·29-s − 1.68·31-s − 1.26·32-s − 1.25·33-s + 0.147·36-s + 1.11·37-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: −1-1
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: 11
Selberg data: (2, 7225, ( :1/2), −1)(2,\ 7225,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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 1−1.88T+2T2 1 - 1.88T + 2T^{2}
3 1−1.88T+3T2 1 - 1.88T + 3T^{2}
7 1−0.335T+7T2 1 - 0.335T + 7T^{2}
11 1+3.80T+11T2 1 + 3.80T + 11T^{2}
13 1−0.558T+13T2 1 - 0.558T + 13T^{2}
19 1+3.73T+19T2 1 + 3.73T + 19T^{2}
23 1−1.24T+23T2 1 - 1.24T + 23T^{2}
29 1+3.98T+29T2 1 + 3.98T + 29T^{2}
31 1+9.36T+31T2 1 + 9.36T + 31T^{2}
37 1−6.77T+37T2 1 - 6.77T + 37T^{2}
41 1−1.68T+41T2 1 - 1.68T + 41T^{2}
43 1+5.66T+43T2 1 + 5.66T + 43T^{2}
47 1−9.50T+47T2 1 - 9.50T + 47T^{2}
53 1−7.30T+53T2 1 - 7.30T + 53T^{2}
59 1+9.33T+59T2 1 + 9.33T + 59T^{2}
61 1+0.743T+61T2 1 + 0.743T + 61T^{2}
67 1+7.29T+67T2 1 + 7.29T + 67T^{2}
71 1+15.5T+71T2 1 + 15.5T + 71T^{2}
73 1−11.1T+73T2 1 - 11.1T + 73T^{2}
79 1−11.9T+79T2 1 - 11.9T + 79T^{2}
83 1+11.7T+83T2 1 + 11.7T + 83T^{2}
89 1−13.1T+89T2 1 - 13.1T + 89T^{2}
97 1−10.6T+97T2 1 - 10.6T + 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

−7.64027052650438020094685611538, −6.81324709434217771930770458624, −5.86644428194017450752063374726, −5.44748645496610974723677577735, −4.58275986487422143192392608800, −3.91124466051000483294378648680, −3.21615478631743197588691421302, −2.57422302469991083216391374153, −1.89924217558148589626514521696, 0, 1.89924217558148589626514521696, 2.57422302469991083216391374153, 3.21615478631743197588691421302, 3.91124466051000483294378648680, 4.58275986487422143192392608800, 5.44748645496610974723677577735, 5.86644428194017450752063374726, 6.81324709434217771930770458624, 7.64027052650438020094685611538

Graph of the ZZ-function along the critical line