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Properties

Label 2-6525-1.1-c1-0-129
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  − 2.74·2-s + 5.52·4-s + 1.10·7-s − 9.68·8-s − 0.0961·11-s − 1.00·13-s − 3.03·14-s + 15.5·16-s − 1.92·17-s + 1.36·19-s + 0.263·22-s − 1.36·23-s + 2.76·26-s + 6.10·28-s + 29-s + 2.17·31-s − 23.1·32-s + 5.28·34-s − 6.61·37-s − 3.75·38-s − 5.07·41-s + 7.53·43-s − 0.531·44-s + 3.74·46-s − 5.77·47-s − 5.78·49-s − 5.57·52-s + ⋯
L(s)  = 1  − 1.94·2-s + 2.76·4-s + 0.417·7-s − 3.42·8-s − 0.0289·11-s − 0.279·13-s − 0.809·14-s + 3.87·16-s − 0.467·17-s + 0.314·19-s + 0.0562·22-s − 0.284·23-s + 0.542·26-s + 1.15·28-s + 0.185·29-s + 0.390·31-s − 4.09·32-s + 0.906·34-s − 1.08·37-s − 0.609·38-s − 0.793·41-s + 1.14·43-s − 0.0801·44-s + 0.552·46-s − 0.843·47-s − 0.825·49-s − 0.772·52-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+2.74T+2T2 1 + 2.74T + 2T^{2}
7 1−1.10T+7T2 1 - 1.10T + 7T^{2}
11 1+0.0961T+11T2 1 + 0.0961T + 11T^{2}
13 1+1.00T+13T2 1 + 1.00T + 13T^{2}
17 1+1.92T+17T2 1 + 1.92T + 17T^{2}
19 1−1.36T+19T2 1 - 1.36T + 19T^{2}
23 1+1.36T+23T2 1 + 1.36T + 23T^{2}
31 1−2.17T+31T2 1 - 2.17T + 31T^{2}
37 1+6.61T+37T2 1 + 6.61T + 37T^{2}
41 1+5.07T+41T2 1 + 5.07T + 41T^{2}
43 1−7.53T+43T2 1 - 7.53T + 43T^{2}
47 1+5.77T+47T2 1 + 5.77T + 47T^{2}
53 1−2.54T+53T2 1 - 2.54T + 53T^{2}
59 1−12.6T+59T2 1 - 12.6T + 59T^{2}
61 1−7.29T+61T2 1 - 7.29T + 61T^{2}
67 1−2.77T+67T2 1 - 2.77T + 67T^{2}
71 1−6.05T+71T2 1 - 6.05T + 71T^{2}
73 1+11.5T+73T2 1 + 11.5T + 73T^{2}
79 1+3.01T+79T2 1 + 3.01T + 79T^{2}
83 1−0.455T+83T2 1 - 0.455T + 83T^{2}
89 1+7.57T+89T2 1 + 7.57T + 89T^{2}
97 1−10.7T+97T2 1 - 10.7T + 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

−7.84483696562995772229409665275, −7.13451706664709801126025684469, −6.66578940633069557172623322128, −5.82174389419070202400229899355, −4.97914938901750851800590969850, −3.72506426892479960409363037273, −2.72527112973358278385329256965, −1.99917436100928333396203774909, −1.12823699302895579248435988605, 0, 1.12823699302895579248435988605, 1.99917436100928333396203774909, 2.72527112973358278385329256965, 3.72506426892479960409363037273, 4.97914938901750851800590969850, 5.82174389419070202400229899355, 6.66578940633069557172623322128, 7.13451706664709801126025684469, 7.84483696562995772229409665275

Graph of the ZZ-function along the critical line