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Properties

Label 2-92-1.1-c5-0-0
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  − 17.0·3-s − 74.6·5-s − 68.6·7-s + 46.5·9-s − 159.·11-s + 193.·13-s + 1.27e3·15-s + 190.·17-s + 1.16e3·19-s + 1.16e3·21-s − 529·23-s + 2.44e3·25-s + 3.34e3·27-s + 3.72e3·29-s + 1.34e3·31-s + 2.70e3·33-s + 5.12e3·35-s − 1.16e3·37-s − 3.30e3·39-s − 1.49e4·41-s + 1.42e4·43-s − 3.47e3·45-s + 1.00e4·47-s − 1.20e4·49-s − 3.24e3·51-s − 2.57e4·53-s + 1.18e4·55-s + ⋯
L(s)  = 1  − 1.09·3-s − 1.33·5-s − 0.529·7-s + 0.191·9-s − 0.396·11-s + 0.318·13-s + 1.45·15-s + 0.160·17-s + 0.737·19-s + 0.578·21-s − 0.208·23-s + 0.783·25-s + 0.882·27-s + 0.822·29-s + 0.251·31-s + 0.432·33-s + 0.707·35-s − 0.140·37-s − 0.347·39-s − 1.38·41-s + 1.17·43-s − 0.255·45-s + 0.664·47-s − 0.719·49-s − 0.174·51-s − 1.26·53-s + 0.529·55-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 0.58316555860.5831655586
L(12)L(\frac12) ≈\approx 0.58316555860.5831655586
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+17.0T+243T2 1 + 17.0T + 243T^{2}
5 1+74.6T+3.12e3T2 1 + 74.6T + 3.12e3T^{2}
7 1+68.6T+1.68e4T2 1 + 68.6T + 1.68e4T^{2}
11 1+159.T+1.61e5T2 1 + 159.T + 1.61e5T^{2}
13 1−193.T+3.71e5T2 1 - 193.T + 3.71e5T^{2}
17 1−190.T+1.41e6T2 1 - 190.T + 1.41e6T^{2}
19 1−1.16e3T+2.47e6T2 1 - 1.16e3T + 2.47e6T^{2}
29 1−3.72e3T+2.05e7T2 1 - 3.72e3T + 2.05e7T^{2}
31 1−1.34e3T+2.86e7T2 1 - 1.34e3T + 2.86e7T^{2}
37 1+1.16e3T+6.93e7T2 1 + 1.16e3T + 6.93e7T^{2}
41 1+1.49e4T+1.15e8T2 1 + 1.49e4T + 1.15e8T^{2}
43 1−1.42e4T+1.47e8T2 1 - 1.42e4T + 1.47e8T^{2}
47 1−1.00e4T+2.29e8T2 1 - 1.00e4T + 2.29e8T^{2}
53 1+2.57e4T+4.18e8T2 1 + 2.57e4T + 4.18e8T^{2}
59 1−967.T+7.14e8T2 1 - 967.T + 7.14e8T^{2}
61 1−2.06e4T+8.44e8T2 1 - 2.06e4T + 8.44e8T^{2}
67 1−1.60e4T+1.35e9T2 1 - 1.60e4T + 1.35e9T^{2}
71 1−3.88e3T+1.80e9T2 1 - 3.88e3T + 1.80e9T^{2}
73 1+4.68e4T+2.07e9T2 1 + 4.68e4T + 2.07e9T^{2}
79 1−2.01e4T+3.07e9T2 1 - 2.01e4T + 3.07e9T^{2}
83 1+3.38e4T+3.93e9T2 1 + 3.38e4T + 3.93e9T^{2}
89 1+5.40e4T+5.58e9T2 1 + 5.40e4T + 5.58e9T^{2}
97 1−9.72e4T+8.58e9T2 1 - 9.72e4T + 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

−12.73490011167862601418878487140, −11.88677630204669728716689607293, −11.20461227232081993215885560704, −10.10682360348157034319629008408, −8.493229421199725669847406581114, −7.32093670214553163835965337794, −6.08762656396737846101137024574, −4.76995473407866603360595538840, −3.31287619356490240476198455428, −0.57517370250827340953864886244, 0.57517370250827340953864886244, 3.31287619356490240476198455428, 4.76995473407866603360595538840, 6.08762656396737846101137024574, 7.32093670214553163835965337794, 8.493229421199725669847406581114, 10.10682360348157034319629008408, 11.20461227232081993215885560704, 11.88677630204669728716689607293, 12.73490011167862601418878487140

Graph of the ZZ-function along the critical line