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Properties

Label 2-74-1.1-c7-0-16
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 + 20.4·3-s + 64·4-s + 375.·5-s − 163.·6-s − 980.·7-s − 512·8-s − 1.76e3·9-s − 3.00e3·10-s − 3.37e3·11-s + 1.31e3·12-s + 3.43e3·13-s + 7.84e3·14-s + 7.69e3·15-s + 4.09e3·16-s − 4.74e3·17-s + 1.41e4·18-s − 1.12e4·19-s + 2.40e4·20-s − 2.01e4·21-s + 2.70e4·22-s − 1.77e4·23-s − 1.04e4·24-s + 6.27e4·25-s − 2.74e4·26-s − 8.10e4·27-s − 6.27e4·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.438·3-s + 0.5·4-s + 1.34·5-s − 0.309·6-s − 1.08·7-s − 0.353·8-s − 0.807·9-s − 0.949·10-s − 0.764·11-s + 0.219·12-s + 0.433·13-s + 0.764·14-s + 0.588·15-s + 0.250·16-s − 0.234·17-s + 0.571·18-s − 0.376·19-s + 0.671·20-s − 0.473·21-s + 0.540·22-s − 0.303·23-s − 0.154·24-s + 0.802·25-s − 0.306·26-s − 0.792·27-s − 0.540·28-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 1−20.4T+2.18e3T2 1 - 20.4T + 2.18e3T^{2}
5 1−375.T+7.81e4T2 1 - 375.T + 7.81e4T^{2}
7 1+980.T+8.23e5T2 1 + 980.T + 8.23e5T^{2}
11 1+3.37e3T+1.94e7T2 1 + 3.37e3T + 1.94e7T^{2}
13 1−3.43e3T+6.27e7T2 1 - 3.43e3T + 6.27e7T^{2}
17 1+4.74e3T+4.10e8T2 1 + 4.74e3T + 4.10e8T^{2}
19 1+1.12e4T+8.93e8T2 1 + 1.12e4T + 8.93e8T^{2}
23 1+1.77e4T+3.40e9T2 1 + 1.77e4T + 3.40e9T^{2}
29 1+1.06e5T+1.72e10T2 1 + 1.06e5T + 1.72e10T^{2}
31 1+3.13e4T+2.75e10T2 1 + 3.13e4T + 2.75e10T^{2}
41 1−2.78e5T+1.94e11T2 1 - 2.78e5T + 1.94e11T^{2}
43 1+6.06e5T+2.71e11T2 1 + 6.06e5T + 2.71e11T^{2}
47 1+1.13e6T+5.06e11T2 1 + 1.13e6T + 5.06e11T^{2}
53 1+8.94e5T+1.17e12T2 1 + 8.94e5T + 1.17e12T^{2}
59 1+8.97e5T+2.48e12T2 1 + 8.97e5T + 2.48e12T^{2}
61 1−1.69e6T+3.14e12T2 1 - 1.69e6T + 3.14e12T^{2}
67 1−2.02e6T+6.06e12T2 1 - 2.02e6T + 6.06e12T^{2}
71 1+5.85e5T+9.09e12T2 1 + 5.85e5T + 9.09e12T^{2}
73 1−4.28e5T+1.10e13T2 1 - 4.28e5T + 1.10e13T^{2}
79 1+1.57e6T+1.92e13T2 1 + 1.57e6T + 1.92e13T^{2}
83 1+5.37e5T+2.71e13T2 1 + 5.37e5T + 2.71e13T^{2}
89 1+1.84e6T+4.42e13T2 1 + 1.84e6T + 4.42e13T^{2}
97 1+6.63e6T+8.07e13T2 1 + 6.63e6T + 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

−12.84750714619757282843393535269, −11.15151075117957220856752637879, −9.989728201688939474581122945223, −9.301337930865519238212383203799, −8.190153499014499833802761633206, −6.55412350511213497195160477471, −5.63118640598797215313043541416, −3.12971676067467901245787159769, −1.99323057322219530211809715948, 0, 1.99323057322219530211809715948, 3.12971676067467901245787159769, 5.63118640598797215313043541416, 6.55412350511213497195160477471, 8.190153499014499833802761633206, 9.301337930865519238212383203799, 9.989728201688939474581122945223, 11.15151075117957220856752637879, 12.84750714619757282843393535269

Graph of the ZZ-function along the critical line