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Properties

Label 2-6525-1.1-c1-0-13
Degree 22
Conductor 65256525
Sign 11
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 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.21·2-s + 2.88·4-s − 1.31·7-s − 1.96·8-s − 4.29·11-s − 2.97·13-s + 2.91·14-s − 1.43·16-s + 0.642·17-s − 5.07·19-s + 9.48·22-s + 8.84·23-s + 6.57·26-s − 3.80·28-s + 29-s − 6.27·31-s + 7.10·32-s − 1.42·34-s + 0.934·37-s + 11.2·38-s − 11.0·41-s + 2.03·43-s − 12.3·44-s − 19.5·46-s + 9.57·47-s − 5.26·49-s − 8.59·52-s + ⋯
L(s)  = 1  − 1.56·2-s + 1.44·4-s − 0.497·7-s − 0.693·8-s − 1.29·11-s − 0.825·13-s + 0.777·14-s − 0.359·16-s + 0.155·17-s − 1.16·19-s + 2.02·22-s + 1.84·23-s + 1.29·26-s − 0.718·28-s + 0.185·29-s − 1.12·31-s + 1.25·32-s − 0.243·34-s + 0.153·37-s + 1.82·38-s − 1.73·41-s + 0.311·43-s − 1.86·44-s − 2.88·46-s + 1.39·47-s − 0.752·49-s − 1.19·52-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: 11
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: 00
Selberg data: (2, 6525, ( :1/2), 1)(2,\ 6525,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.31083013280.3108301328
L(12)L(\frac12) ≈\approx 0.31083013280.3108301328
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+2.21T+2T2 1 + 2.21T + 2T^{2}
7 1+1.31T+7T2 1 + 1.31T + 7T^{2}
11 1+4.29T+11T2 1 + 4.29T + 11T^{2}
13 1+2.97T+13T2 1 + 2.97T + 13T^{2}
17 1−0.642T+17T2 1 - 0.642T + 17T^{2}
19 1+5.07T+19T2 1 + 5.07T + 19T^{2}
23 1−8.84T+23T2 1 - 8.84T + 23T^{2}
31 1+6.27T+31T2 1 + 6.27T + 31T^{2}
37 1−0.934T+37T2 1 - 0.934T + 37T^{2}
41 1+11.0T+41T2 1 + 11.0T + 41T^{2}
43 1−2.03T+43T2 1 - 2.03T + 43T^{2}
47 1−9.57T+47T2 1 - 9.57T + 47T^{2}
53 1+13.0T+53T2 1 + 13.0T + 53T^{2}
59 1−2.55T+59T2 1 - 2.55T + 59T^{2}
61 1−8.46T+61T2 1 - 8.46T + 61T^{2}
67 1+12.9T+67T2 1 + 12.9T + 67T^{2}
71 1+5.10T+71T2 1 + 5.10T + 71T^{2}
73 1+16.0T+73T2 1 + 16.0T + 73T^{2}
79 1+3.30T+79T2 1 + 3.30T + 79T^{2}
83 1−5.24T+83T2 1 - 5.24T + 83T^{2}
89 1−5.05T+89T2 1 - 5.05T + 89T^{2}
97 1+2.04T+97T2 1 + 2.04T + 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

−8.084214920528082389992513022662, −7.38185137562722214846042288431, −7.01089278134334667569510429605, −6.15730940538852303152032400188, −5.19334302910783722460392386525, −4.53871982187785527874185606198, −3.19445895249833723337321642318, −2.53460544967714176890197326784, −1.64012778103215486149588899014, −0.35654533144548996928599311603, 0.35654533144548996928599311603, 1.64012778103215486149588899014, 2.53460544967714176890197326784, 3.19445895249833723337321642318, 4.53871982187785527874185606198, 5.19334302910783722460392386525, 6.15730940538852303152032400188, 7.01089278134334667569510429605, 7.38185137562722214846042288431, 8.084214920528082389992513022662

Graph of the ZZ-function along the critical line