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Properties

Label 2-74-1.1-c7-0-18
Degree 22
Conductor 7474
Sign −1-1
Analytic cond. 23.116423.1164
Root an. cond. 4.807964.80796
Motivic weight 77
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 8·2-s + 22.3·3-s + 64·4-s − 504.·5-s + 179.·6-s + 892.·7-s + 512·8-s − 1.68e3·9-s − 4.03e3·10-s + 4.70e3·11-s + 1.43e3·12-s − 1.18e4·13-s + 7.13e3·14-s − 1.12e4·15-s + 4.09e3·16-s − 2.26e4·17-s − 1.34e4·18-s − 5.16e4·19-s − 3.22e4·20-s + 1.99e4·21-s + 3.76e4·22-s − 7.28e4·23-s + 1.14e4·24-s + 1.76e5·25-s − 9.47e4·26-s − 8.67e4·27-s + 5.71e4·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.478·3-s + 0.5·4-s − 1.80·5-s + 0.338·6-s + 0.983·7-s + 0.353·8-s − 0.770·9-s − 1.27·10-s + 1.06·11-s + 0.239·12-s − 1.49·13-s + 0.695·14-s − 0.864·15-s + 0.250·16-s − 1.11·17-s − 0.545·18-s − 1.72·19-s − 0.902·20-s + 0.470·21-s + 0.753·22-s − 1.24·23-s + 0.169·24-s + 2.25·25-s − 1.05·26-s − 0.847·27-s + 0.491·28-s + ⋯

Functional equation

Λ(s)=(74s/2 ΓC(s) L(s)=(− Λ(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}
Λ(s)=(74s/2 ΓC(s+7/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 7474    =    2⋅372 \cdot 37
Sign: −1-1
Analytic conductor: 23.116423.1164
Root analytic conductor: 4.807964.80796
Motivic weight: 77
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 74, ( :7/2), −1)(2,\ 74,\ (\ :7/2),\ -1)

Particular Values

L(4)L(4) == 00
L(12)L(\frac12) == 00
L(92)L(\frac{9}{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 1−8T 1 - 8T
37 1+5.06e4T 1 + 5.06e4T
good3 1−22.3T+2.18e3T2 1 - 22.3T + 2.18e3T^{2}
5 1+504.T+7.81e4T2 1 + 504.T + 7.81e4T^{2}
7 1−892.T+8.23e5T2 1 - 892.T + 8.23e5T^{2}
11 1−4.70e3T+1.94e7T2 1 - 4.70e3T + 1.94e7T^{2}
13 1+1.18e4T+6.27e7T2 1 + 1.18e4T + 6.27e7T^{2}
17 1+2.26e4T+4.10e8T2 1 + 2.26e4T + 4.10e8T^{2}
19 1+5.16e4T+8.93e8T2 1 + 5.16e4T + 8.93e8T^{2}
23 1+7.28e4T+3.40e9T2 1 + 7.28e4T + 3.40e9T^{2}
29 1−1.62e5T+1.72e10T2 1 - 1.62e5T + 1.72e10T^{2}
31 1+2.03e4T+2.75e10T2 1 + 2.03e4T + 2.75e10T^{2}
41 1+5.40e5T+1.94e11T2 1 + 5.40e5T + 1.94e11T^{2}
43 1−1.58e5T+2.71e11T2 1 - 1.58e5T + 2.71e11T^{2}
47 1−4.70e5T+5.06e11T2 1 - 4.70e5T + 5.06e11T^{2}
53 1−1.52e6T+1.17e12T2 1 - 1.52e6T + 1.17e12T^{2}
59 1−1.33e6T+2.48e12T2 1 - 1.33e6T + 2.48e12T^{2}
61 1+3.08e6T+3.14e12T2 1 + 3.08e6T + 3.14e12T^{2}
67 1−2.02e5T+6.06e12T2 1 - 2.02e5T + 6.06e12T^{2}
71 1+1.68e6T+9.09e12T2 1 + 1.68e6T + 9.09e12T^{2}
73 1−3.90e6T+1.10e13T2 1 - 3.90e6T + 1.10e13T^{2}
79 1−4.25e6T+1.92e13T2 1 - 4.25e6T + 1.92e13T^{2}
83 1−2.69e6T+2.71e13T2 1 - 2.69e6T + 2.71e13T^{2}
89 1−6.50e6T+4.42e13T2 1 - 6.50e6T + 4.42e13T^{2}
97 1+7.24e6T+8.07e13T2 1 + 7.24e6T + 8.07e13T^{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

−12.22708014114808240083355943309, −11.79871004670638855411458284575, −10.79091518162117528639326852917, −8.700110102557052583373379688522, −7.948327331037264779109150561429, −6.72621037937633725945670666169, −4.68917329768906413932991237955, −3.93188502286257509403606520755, −2.32172408316003384122020943012, 0, 2.32172408316003384122020943012, 3.93188502286257509403606520755, 4.68917329768906413932991237955, 6.72621037937633725945670666169, 7.948327331037264779109150561429, 8.700110102557052583373379688522, 10.79091518162117528639326852917, 11.79871004670638855411458284575, 12.22708014114808240083355943309

Graph of the ZZ-function along the critical line