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Properties

Label 2-6525-1.1-c1-0-136
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  + 0.747·2-s − 1.44·4-s − 3.28·7-s − 2.57·8-s + 1.14·11-s + 2.14·13-s − 2.45·14-s + 0.958·16-s + 6.16·17-s − 7.20·19-s + 0.853·22-s − 0.227·23-s + 1.60·26-s + 4.73·28-s + 29-s + 1.59·31-s + 5.86·32-s + 4.60·34-s + 0.690·37-s − 5.38·38-s + 9.55·41-s + 0.739·43-s − 1.64·44-s − 0.170·46-s − 3.52·47-s + 3.80·49-s − 3.09·52-s + ⋯
L(s)  = 1  + 0.528·2-s − 0.720·4-s − 1.24·7-s − 0.909·8-s + 0.344·11-s + 0.595·13-s − 0.656·14-s + 0.239·16-s + 1.49·17-s − 1.65·19-s + 0.181·22-s − 0.0474·23-s + 0.314·26-s + 0.895·28-s + 0.185·29-s + 0.285·31-s + 1.03·32-s + 0.790·34-s + 0.113·37-s − 0.873·38-s + 1.49·41-s + 0.112·43-s − 0.248·44-s − 0.0250·46-s − 0.514·47-s + 0.544·49-s − 0.428·52-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 1−0.747T+2T2 1 - 0.747T + 2T^{2}
7 1+3.28T+7T2 1 + 3.28T + 7T^{2}
11 1−1.14T+11T2 1 - 1.14T + 11T^{2}
13 1−2.14T+13T2 1 - 2.14T + 13T^{2}
17 1−6.16T+17T2 1 - 6.16T + 17T^{2}
19 1+7.20T+19T2 1 + 7.20T + 19T^{2}
23 1+0.227T+23T2 1 + 0.227T + 23T^{2}
31 1−1.59T+31T2 1 - 1.59T + 31T^{2}
37 1−0.690T+37T2 1 - 0.690T + 37T^{2}
41 1−9.55T+41T2 1 - 9.55T + 41T^{2}
43 1−0.739T+43T2 1 - 0.739T + 43T^{2}
47 1+3.52T+47T2 1 + 3.52T + 47T^{2}
53 1−0.756T+53T2 1 - 0.756T + 53T^{2}
59 1−7.99T+59T2 1 - 7.99T + 59T^{2}
61 1+0.836T+61T2 1 + 0.836T + 61T^{2}
67 1+6.00T+67T2 1 + 6.00T + 67T^{2}
71 1+10.8T+71T2 1 + 10.8T + 71T^{2}
73 1+6.33T+73T2 1 + 6.33T + 73T^{2}
79 1+7.23T+79T2 1 + 7.23T + 79T^{2}
83 1+7.46T+83T2 1 + 7.46T + 83T^{2}
89 1+0.620T+89T2 1 + 0.620T + 89T^{2}
97 1+4.33T+97T2 1 + 4.33T + 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

−7.66355292533562620356126899397, −6.73437690482299884612221934611, −6.02068729912820264752844057604, −5.74714510830185429567556144276, −4.61660711004747213935838485981, −3.98802193718855203790360650028, −3.36303556407738172638295172926, −2.62820325589259683832266208545, −1.15571206242245060112864506073, 0, 1.15571206242245060112864506073, 2.62820325589259683832266208545, 3.36303556407738172638295172926, 3.98802193718855203790360650028, 4.61660711004747213935838485981, 5.74714510830185429567556144276, 6.02068729912820264752844057604, 6.73437690482299884612221934611, 7.66355292533562620356126899397

Graph of the ZZ-function along the critical line