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Properties

Label 2-92-1.1-c5-0-2
Degree 22
Conductor 9292
Sign 11
Analytic cond. 14.755314.7553
Root an. cond. 3.841263.84126
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.77·3-s + 80.6·5-s + 236.·7-s − 239.·9-s − 634.·11-s + 374.·13-s + 142.·15-s + 990.·17-s + 2.96e3·19-s + 419.·21-s − 529·23-s + 3.38e3·25-s − 854.·27-s − 784.·29-s + 3.77e3·31-s − 1.12e3·33-s + 1.90e4·35-s − 4.50e3·37-s + 662.·39-s − 7.58e3·41-s + 6.51e3·43-s − 1.93e4·45-s − 8.90e3·47-s + 3.92e4·49-s + 1.75e3·51-s + 1.17e3·53-s − 5.12e4·55-s + ⋯
L(s)  = 1  + 0.113·3-s + 1.44·5-s + 1.82·7-s − 0.987·9-s − 1.58·11-s + 0.614·13-s + 0.163·15-s + 0.831·17-s + 1.88·19-s + 0.207·21-s − 0.208·23-s + 1.08·25-s − 0.225·27-s − 0.173·29-s + 0.706·31-s − 0.179·33-s + 2.63·35-s − 0.541·37-s + 0.0697·39-s − 0.705·41-s + 0.537·43-s − 1.42·45-s − 0.587·47-s + 2.33·49-s + 0.0943·51-s + 0.0576·53-s − 2.28·55-s + ⋯

Functional equation

Λ(s)=(92s/2 ΓC(s) L(s)=( Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(92s/2 ΓC(s+5/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 9292    =    22⋅232^{2} \cdot 23
Sign: 11
Analytic conductor: 14.755314.7553
Root analytic conductor: 3.841263.84126
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 92, ( :5/2), 1)(2,\ 92,\ (\ :5/2),\ 1)

Particular Values

L(3)L(3) ≈\approx 2.6650912402.665091240
L(12)L(\frac12) ≈\approx 2.6650912402.665091240
L(72)L(\frac{7}{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)
bad2 1 1
23 1+529T 1 + 529T
good3 1−1.77T+243T2 1 - 1.77T + 243T^{2}
5 1−80.6T+3.12e3T2 1 - 80.6T + 3.12e3T^{2}
7 1−236.T+1.68e4T2 1 - 236.T + 1.68e4T^{2}
11 1+634.T+1.61e5T2 1 + 634.T + 1.61e5T^{2}
13 1−374.T+3.71e5T2 1 - 374.T + 3.71e5T^{2}
17 1−990.T+1.41e6T2 1 - 990.T + 1.41e6T^{2}
19 1−2.96e3T+2.47e6T2 1 - 2.96e3T + 2.47e6T^{2}
29 1+784.T+2.05e7T2 1 + 784.T + 2.05e7T^{2}
31 1−3.77e3T+2.86e7T2 1 - 3.77e3T + 2.86e7T^{2}
37 1+4.50e3T+6.93e7T2 1 + 4.50e3T + 6.93e7T^{2}
41 1+7.58e3T+1.15e8T2 1 + 7.58e3T + 1.15e8T^{2}
43 1−6.51e3T+1.47e8T2 1 - 6.51e3T + 1.47e8T^{2}
47 1+8.90e3T+2.29e8T2 1 + 8.90e3T + 2.29e8T^{2}
53 1−1.17e3T+4.18e8T2 1 - 1.17e3T + 4.18e8T^{2}
59 1−3.88e4T+7.14e8T2 1 - 3.88e4T + 7.14e8T^{2}
61 1+2.21e4T+8.44e8T2 1 + 2.21e4T + 8.44e8T^{2}
67 1+4.42e4T+1.35e9T2 1 + 4.42e4T + 1.35e9T^{2}
71 1−3.13e4T+1.80e9T2 1 - 3.13e4T + 1.80e9T^{2}
73 1+7.74e4T+2.07e9T2 1 + 7.74e4T + 2.07e9T^{2}
79 1+1.60e4T+3.07e9T2 1 + 1.60e4T + 3.07e9T^{2}
83 1−2.79e4T+3.93e9T2 1 - 2.79e4T + 3.93e9T^{2}
89 1+1.06e5T+5.58e9T2 1 + 1.06e5T + 5.58e9T^{2}
97 1+7.12e4T+8.58e9T2 1 + 7.12e4T + 8.58e9T^{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

−13.55721467313517134751474027832, −11.89472478067585804108828950234, −10.89543169958750374650069618905, −9.912314601653357142722744074735, −8.539285334484289554159995492114, −7.67978816795963632942594731031, −5.61826898647124550696559629666, −5.21339087819810365207902315072, −2.75515942074681724524511972563, −1.41912085942650671140141303815, 1.41912085942650671140141303815, 2.75515942074681724524511972563, 5.21339087819810365207902315072, 5.61826898647124550696559629666, 7.67978816795963632942594731031, 8.539285334484289554159995492114, 9.912314601653357142722744074735, 10.89543169958750374650069618905, 11.89472478067585804108828950234, 13.55721467313517134751474027832

Graph of the ZZ-function along the critical line