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Properties

Label 2-6525-1.1-c1-0-206
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  + 2.07·2-s + 2.32·4-s − 0.602·7-s + 0.670·8-s + 3.75·11-s − 3.15·13-s − 1.25·14-s − 3.25·16-s − 5.57·17-s − 3.67·19-s + 7.80·22-s + 5.90·23-s − 6.54·26-s − 1.39·28-s + 29-s − 3.09·31-s − 8.09·32-s − 11.5·34-s − 9.93·37-s − 7.63·38-s + 0.0283·41-s + 5.05·43-s + 8.71·44-s + 12.2·46-s − 2.15·47-s − 6.63·49-s − 7.31·52-s + ⋯
L(s)  = 1  + 1.47·2-s + 1.16·4-s − 0.227·7-s + 0.237·8-s + 1.13·11-s − 0.873·13-s − 0.334·14-s − 0.812·16-s − 1.35·17-s − 0.842·19-s + 1.66·22-s + 1.23·23-s − 1.28·26-s − 0.264·28-s + 0.185·29-s − 0.555·31-s − 1.43·32-s − 1.98·34-s − 1.63·37-s − 1.23·38-s + 0.00443·41-s + 0.770·43-s + 1.31·44-s + 1.81·46-s − 0.314·47-s − 0.948·49-s − 1.01·52-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 1−2.07T+2T2 1 - 2.07T + 2T^{2}
7 1+0.602T+7T2 1 + 0.602T + 7T^{2}
11 1−3.75T+11T2 1 - 3.75T + 11T^{2}
13 1+3.15T+13T2 1 + 3.15T + 13T^{2}
17 1+5.57T+17T2 1 + 5.57T + 17T^{2}
19 1+3.67T+19T2 1 + 3.67T + 19T^{2}
23 1−5.90T+23T2 1 - 5.90T + 23T^{2}
31 1+3.09T+31T2 1 + 3.09T + 31T^{2}
37 1+9.93T+37T2 1 + 9.93T + 37T^{2}
41 1−0.0283T+41T2 1 - 0.0283T + 41T^{2}
43 1−5.05T+43T2 1 - 5.05T + 43T^{2}
47 1+2.15T+47T2 1 + 2.15T + 47T^{2}
53 1+11.6T+53T2 1 + 11.6T + 53T^{2}
59 1−9.99T+59T2 1 - 9.99T + 59T^{2}
61 1+5.22T+61T2 1 + 5.22T + 61T^{2}
67 1+12.7T+67T2 1 + 12.7T + 67T^{2}
71 1−5.71T+71T2 1 - 5.71T + 71T^{2}
73 1−8.39T+73T2 1 - 8.39T + 73T^{2}
79 1−9.33T+79T2 1 - 9.33T + 79T^{2}
83 1+13.6T+83T2 1 + 13.6T + 83T^{2}
89 1−10.8T+89T2 1 - 10.8T + 89T^{2}
97 1+11.3T+97T2 1 + 11.3T + 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

−7.20993138234823062778039892090, −6.69753410188799093729545654694, −6.30188263788654961924359227320, −5.30607853523701675203518630657, −4.72399464498383652840647426468, −4.12243707137649188093188200487, −3.37965066541536046710775592995, −2.56961685036824597309077214751, −1.70349115438103068567087627505, 0, 1.70349115438103068567087627505, 2.56961685036824597309077214751, 3.37965066541536046710775592995, 4.12243707137649188093188200487, 4.72399464498383652840647426468, 5.30607853523701675203518630657, 6.30188263788654961924359227320, 6.69753410188799093729545654694, 7.20993138234823062778039892090

Graph of the ZZ-function along the critical line