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Properties

Label 2-538-1.1-c7-0-24
Degree 22
Conductor 538538
Sign 11
Analytic cond. 168.063168.063
Root an. cond. 12.963912.9639
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 + 2.39·3-s + 64·4-s − 162.·5-s − 19.1·6-s + 677.·7-s − 512·8-s − 2.18e3·9-s + 1.29e3·10-s − 5.83e3·11-s + 153.·12-s + 7.76e3·13-s − 5.42e3·14-s − 388.·15-s + 4.09e3·16-s + 3.05e4·17-s + 1.74e4·18-s − 2.00e4·19-s − 1.03e4·20-s + 1.62e3·21-s + 4.67e4·22-s − 2.06e4·23-s − 1.22e3·24-s − 5.18e4·25-s − 6.21e4·26-s − 1.04e4·27-s + 4.33e4·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.0512·3-s + 0.5·4-s − 0.580·5-s − 0.0362·6-s + 0.746·7-s − 0.353·8-s − 0.997·9-s + 0.410·10-s − 1.32·11-s + 0.0256·12-s + 0.980·13-s − 0.528·14-s − 0.0297·15-s + 0.250·16-s + 1.50·17-s + 0.705·18-s − 0.670·19-s − 0.290·20-s + 0.0382·21-s + 0.935·22-s − 0.354·23-s − 0.0181·24-s − 0.663·25-s − 0.693·26-s − 0.102·27-s + 0.373·28-s + ⋯

Functional equation

Λ(s)=(538s/2 ΓC(s) L(s)=( Λ(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}
Λ(s)=(538s/2 ΓC(s+7/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 538538    =    2⋅2692 \cdot 269
Sign: 11
Analytic conductor: 168.063168.063
Root analytic conductor: 12.963912.9639
Motivic weight: 77
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 538, ( :7/2), 1)(2,\ 538,\ (\ :7/2),\ 1)

Particular Values

L(4)L(4) ≈\approx 0.88180370000.8818037000
L(12)L(\frac12) ≈\approx 0.88180370000.8818037000
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
269 1+1.94e7T 1 + 1.94e7T
good3 1−2.39T+2.18e3T2 1 - 2.39T + 2.18e3T^{2}
5 1+162.T+7.81e4T2 1 + 162.T + 7.81e4T^{2}
7 1−677.T+8.23e5T2 1 - 677.T + 8.23e5T^{2}
11 1+5.83e3T+1.94e7T2 1 + 5.83e3T + 1.94e7T^{2}
13 1−7.76e3T+6.27e7T2 1 - 7.76e3T + 6.27e7T^{2}
17 1−3.05e4T+4.10e8T2 1 - 3.05e4T + 4.10e8T^{2}
19 1+2.00e4T+8.93e8T2 1 + 2.00e4T + 8.93e8T^{2}
23 1+2.06e4T+3.40e9T2 1 + 2.06e4T + 3.40e9T^{2}
29 1−6.42e4T+1.72e10T2 1 - 6.42e4T + 1.72e10T^{2}
31 1+3.01e5T+2.75e10T2 1 + 3.01e5T + 2.75e10T^{2}
37 1−1.49e5T+9.49e10T2 1 - 1.49e5T + 9.49e10T^{2}
41 1+1.98e5T+1.94e11T2 1 + 1.98e5T + 1.94e11T^{2}
43 1−1.41e5T+2.71e11T2 1 - 1.41e5T + 2.71e11T^{2}
47 1−7.38e5T+5.06e11T2 1 - 7.38e5T + 5.06e11T^{2}
53 1+6.58e5T+1.17e12T2 1 + 6.58e5T + 1.17e12T^{2}
59 1+3.05e5T+2.48e12T2 1 + 3.05e5T + 2.48e12T^{2}
61 1+1.52e5T+3.14e12T2 1 + 1.52e5T + 3.14e12T^{2}
67 1−3.75e6T+6.06e12T2 1 - 3.75e6T + 6.06e12T^{2}
71 1−2.85e6T+9.09e12T2 1 - 2.85e6T + 9.09e12T^{2}
73 1+4.51e6T+1.10e13T2 1 + 4.51e6T + 1.10e13T^{2}
79 1+3.43e4T+1.92e13T2 1 + 3.43e4T + 1.92e13T^{2}
83 1+8.98e6T+2.71e13T2 1 + 8.98e6T + 2.71e13T^{2}
89 1−2.09e6T+4.42e13T2 1 - 2.09e6T + 4.42e13T^{2}
97 1+1.21e7T+8.07e13T2 1 + 1.21e7T + 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

−9.664496878214983444223585741957, −8.468544916315708045165602945354, −8.115123954848724560883015174236, −7.39682058776330682510369912118, −5.95295961416069673424933696136, −5.29865121189650288668412628038, −3.84797018895690369928062816468, −2.83663677001601010039740271723, −1.69065033380868536861233220455, −0.44728684271588943537623775702, 0.44728684271588943537623775702, 1.69065033380868536861233220455, 2.83663677001601010039740271723, 3.84797018895690369928062816468, 5.29865121189650288668412628038, 5.95295961416069673424933696136, 7.39682058776330682510369912118, 8.115123954848724560883015174236, 8.468544916315708045165602945354, 9.664496878214983444223585741957

Graph of the ZZ-function along the critical line