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Properties

Label 2-6525-1.1-c1-0-100
Degree 22
Conductor 65256525
Sign −1-1
Analytic cond. 52.102352.1023
Root an. cond. 7.218197.21819
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  − 0.712·2-s − 1.49·4-s − 2.77·7-s + 2.48·8-s − 4.26·11-s + 0.779·13-s + 1.98·14-s + 1.21·16-s + 1.90·17-s + 6.72·19-s + 3.04·22-s − 2.17·23-s − 0.555·26-s + 4.14·28-s − 29-s − 8.82·31-s − 5.83·32-s − 1.35·34-s + 1.48·37-s − 4.78·38-s + 7.71·41-s − 8.19·43-s + 6.36·44-s + 1.54·46-s + 5.19·47-s + 0.727·49-s − 1.16·52-s + ⋯
L(s)  = 1  − 0.503·2-s − 0.746·4-s − 1.05·7-s + 0.879·8-s − 1.28·11-s + 0.216·13-s + 0.529·14-s + 0.302·16-s + 0.461·17-s + 1.54·19-s + 0.648·22-s − 0.452·23-s − 0.108·26-s + 0.783·28-s − 0.185·29-s − 1.58·31-s − 1.03·32-s − 0.232·34-s + 0.243·37-s − 0.776·38-s + 1.20·41-s − 1.24·43-s + 0.960·44-s + 0.228·46-s + 0.757·47-s + 0.103·49-s − 0.161·52-s + ⋯

Functional equation

Λ(s)=(6525s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 6525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(6525s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 6525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 65256525    =    32⋅52⋅293^{2} \cdot 5^{2} \cdot 29
Sign: −1-1
Analytic conductor: 52.102352.1023
Root analytic conductor: 7.218197.21819
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 6525, ( :1/2), −1)(2,\ 6525,\ (\ :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)
bad3 1 1
5 1 1
29 1+T 1 + T
good2 1+0.712T+2T2 1 + 0.712T + 2T^{2}
7 1+2.77T+7T2 1 + 2.77T + 7T^{2}
11 1+4.26T+11T2 1 + 4.26T + 11T^{2}
13 1−0.779T+13T2 1 - 0.779T + 13T^{2}
17 1−1.90T+17T2 1 - 1.90T + 17T^{2}
19 1−6.72T+19T2 1 - 6.72T + 19T^{2}
23 1+2.17T+23T2 1 + 2.17T + 23T^{2}
31 1+8.82T+31T2 1 + 8.82T + 31T^{2}
37 1−1.48T+37T2 1 - 1.48T + 37T^{2}
41 1−7.71T+41T2 1 - 7.71T + 41T^{2}
43 1+8.19T+43T2 1 + 8.19T + 43T^{2}
47 1−5.19T+47T2 1 - 5.19T + 47T^{2}
53 1−11.7T+53T2 1 - 11.7T + 53T^{2}
59 1−4.46T+59T2 1 - 4.46T + 59T^{2}
61 1+5.24T+61T2 1 + 5.24T + 61T^{2}
67 1+8.49T+67T2 1 + 8.49T + 67T^{2}
71 1+0.663T+71T2 1 + 0.663T + 71T^{2}
73 1−16.5T+73T2 1 - 16.5T + 73T^{2}
79 1−9.54T+79T2 1 - 9.54T + 79T^{2}
83 1−0.0123T+83T2 1 - 0.0123T + 83T^{2}
89 1−5.46T+89T2 1 - 5.46T + 89T^{2}
97 1+0.952T+97T2 1 + 0.952T + 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

−7.51998321050715498695386334560, −7.39557779205627741289473115590, −6.15778430321575164439314904428, −5.46293907034781336450100306864, −4.96477622157978970111524557238, −3.80505567277911078955317864042, −3.32132806073536577773722155801, −2.28426918983887133514478014970, −0.982930349876388716950656143583, 0, 0.982930349876388716950656143583, 2.28426918983887133514478014970, 3.32132806073536577773722155801, 3.80505567277911078955317864042, 4.96477622157978970111524557238, 5.46293907034781336450100306864, 6.15778430321575164439314904428, 7.39557779205627741289473115590, 7.51998321050715498695386334560

Graph of the ZZ-function along the critical line