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Properties

Label 2-74-1.1-c7-0-8
Degree 22
Conductor 7474
Sign 11
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 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·2-s + 62.0·3-s + 64·4-s + 42.8·5-s − 496.·6-s + 819.·7-s − 512·8-s + 1.65e3·9-s − 342.·10-s + 728.·11-s + 3.96e3·12-s + 1.43e4·13-s − 6.55e3·14-s + 2.65e3·15-s + 4.09e3·16-s − 3.48e4·17-s − 1.32e4·18-s + 3.87e4·19-s + 2.74e3·20-s + 5.08e4·21-s − 5.83e3·22-s − 6.14e4·23-s − 3.17e4·24-s − 7.62e4·25-s − 1.14e5·26-s − 3.27e4·27-s + 5.24e4·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.32·3-s + 0.5·4-s + 0.153·5-s − 0.937·6-s + 0.902·7-s − 0.353·8-s + 0.758·9-s − 0.108·10-s + 0.165·11-s + 0.663·12-s + 1.80·13-s − 0.638·14-s + 0.203·15-s + 0.250·16-s − 1.72·17-s − 0.536·18-s + 1.29·19-s + 0.0766·20-s + 1.19·21-s − 0.116·22-s − 1.05·23-s − 0.468·24-s − 0.976·25-s − 1.27·26-s − 0.319·27-s + 0.451·28-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: 11
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: 00
Selberg data: (2, 74, ( :7/2), 1)(2,\ 74,\ (\ :7/2),\ 1)

Particular Values

L(4)L(4) ≈\approx 2.6324411482.632441148
L(12)L(\frac12) ≈\approx 2.6324411482.632441148
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 1−62.0T+2.18e3T2 1 - 62.0T + 2.18e3T^{2}
5 1−42.8T+7.81e4T2 1 - 42.8T + 7.81e4T^{2}
7 1−819.T+8.23e5T2 1 - 819.T + 8.23e5T^{2}
11 1−728.T+1.94e7T2 1 - 728.T + 1.94e7T^{2}
13 1−1.43e4T+6.27e7T2 1 - 1.43e4T + 6.27e7T^{2}
17 1+3.48e4T+4.10e8T2 1 + 3.48e4T + 4.10e8T^{2}
19 1−3.87e4T+8.93e8T2 1 - 3.87e4T + 8.93e8T^{2}
23 1+6.14e4T+3.40e9T2 1 + 6.14e4T + 3.40e9T^{2}
29 1−1.97e5T+1.72e10T2 1 - 1.97e5T + 1.72e10T^{2}
31 1−2.55e5T+2.75e10T2 1 - 2.55e5T + 2.75e10T^{2}
41 1−7.97e5T+1.94e11T2 1 - 7.97e5T + 1.94e11T^{2}
43 1−1.41e5T+2.71e11T2 1 - 1.41e5T + 2.71e11T^{2}
47 1−6.43e5T+5.06e11T2 1 - 6.43e5T + 5.06e11T^{2}
53 1−9.94e5T+1.17e12T2 1 - 9.94e5T + 1.17e12T^{2}
59 1+2.48e6T+2.48e12T2 1 + 2.48e6T + 2.48e12T^{2}
61 1+6.49e5T+3.14e12T2 1 + 6.49e5T + 3.14e12T^{2}
67 1+8.18e5T+6.06e12T2 1 + 8.18e5T + 6.06e12T^{2}
71 1+2.77e6T+9.09e12T2 1 + 2.77e6T + 9.09e12T^{2}
73 1−4.46e6T+1.10e13T2 1 - 4.46e6T + 1.10e13T^{2}
79 1−3.61e6T+1.92e13T2 1 - 3.61e6T + 1.92e13T^{2}
83 1+2.47e6T+2.71e13T2 1 + 2.47e6T + 2.71e13T^{2}
89 1−6.02e6T+4.42e13T2 1 - 6.02e6T + 4.42e13T^{2}
97 1+1.02e7T+8.07e13T2 1 + 1.02e7T + 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

−13.67777415214370083076010595469, −11.79244311522966693660340573043, −10.76471576410348553482768296803, −9.377339910894864726580943638083, −8.517306159320857213464598726570, −7.83122459663564787031766662487, −6.21661975984554573517147321830, −4.10187065682825513668347355537, −2.54499302609837275849153217123, −1.28407571223361304310102957593, 1.28407571223361304310102957593, 2.54499302609837275849153217123, 4.10187065682825513668347355537, 6.21661975984554573517147321830, 7.83122459663564787031766662487, 8.517306159320857213464598726570, 9.377339910894864726580943638083, 10.76471576410348553482768296803, 11.79244311522966693660340573043, 13.67777415214370083076010595469

Graph of the ZZ-function along the critical line