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Properties

Label 2-538-1.1-c7-0-40
Degree 22
Conductor 538538
Sign −1-1
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 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·2-s − 82.6·3-s + 64·4-s − 228.·5-s + 661.·6-s − 449.·7-s − 512·8-s + 4.64e3·9-s + 1.82e3·10-s − 1.32e3·11-s − 5.29e3·12-s − 8.21e3·13-s + 3.59e3·14-s + 1.88e4·15-s + 4.09e3·16-s + 2.25e4·17-s − 3.71e4·18-s − 2.92e4·19-s − 1.46e4·20-s + 3.71e4·21-s + 1.05e4·22-s − 9.67e4·23-s + 4.23e4·24-s − 2.60e4·25-s + 6.57e4·26-s − 2.03e5·27-s − 2.87e4·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.76·3-s + 0.5·4-s − 0.816·5-s + 1.25·6-s − 0.495·7-s − 0.353·8-s + 2.12·9-s + 0.577·10-s − 0.299·11-s − 0.883·12-s − 1.03·13-s + 0.350·14-s + 1.44·15-s + 0.250·16-s + 1.11·17-s − 1.50·18-s − 0.978·19-s − 0.408·20-s + 0.876·21-s + 0.211·22-s − 1.65·23-s + 0.625·24-s − 0.333·25-s + 0.733·26-s − 1.98·27-s − 0.247·28-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: −1-1
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: 11
Selberg data: (2, 538, ( :7/2), −1)(2,\ 538,\ (\ :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
269 1−1.94e7T 1 - 1.94e7T
good3 1+82.6T+2.18e3T2 1 + 82.6T + 2.18e3T^{2}
5 1+228.T+7.81e4T2 1 + 228.T + 7.81e4T^{2}
7 1+449.T+8.23e5T2 1 + 449.T + 8.23e5T^{2}
11 1+1.32e3T+1.94e7T2 1 + 1.32e3T + 1.94e7T^{2}
13 1+8.21e3T+6.27e7T2 1 + 8.21e3T + 6.27e7T^{2}
17 1−2.25e4T+4.10e8T2 1 - 2.25e4T + 4.10e8T^{2}
19 1+2.92e4T+8.93e8T2 1 + 2.92e4T + 8.93e8T^{2}
23 1+9.67e4T+3.40e9T2 1 + 9.67e4T + 3.40e9T^{2}
29 1−1.38e5T+1.72e10T2 1 - 1.38e5T + 1.72e10T^{2}
31 1+2.60e5T+2.75e10T2 1 + 2.60e5T + 2.75e10T^{2}
37 1−8.25e4T+9.49e10T2 1 - 8.25e4T + 9.49e10T^{2}
41 1−7.88e5T+1.94e11T2 1 - 7.88e5T + 1.94e11T^{2}
43 1+9.96e5T+2.71e11T2 1 + 9.96e5T + 2.71e11T^{2}
47 1+7.62e4T+5.06e11T2 1 + 7.62e4T + 5.06e11T^{2}
53 1−4.00e5T+1.17e12T2 1 - 4.00e5T + 1.17e12T^{2}
59 1−1.76e6T+2.48e12T2 1 - 1.76e6T + 2.48e12T^{2}
61 1+2.45e6T+3.14e12T2 1 + 2.45e6T + 3.14e12T^{2}
67 1−8.69e5T+6.06e12T2 1 - 8.69e5T + 6.06e12T^{2}
71 1−1.65e6T+9.09e12T2 1 - 1.65e6T + 9.09e12T^{2}
73 1−5.17e6T+1.10e13T2 1 - 5.17e6T + 1.10e13T^{2}
79 1−6.08e6T+1.92e13T2 1 - 6.08e6T + 1.92e13T^{2}
83 1+6.19e6T+2.71e13T2 1 + 6.19e6T + 2.71e13T^{2}
89 1−5.65e6T+4.42e13T2 1 - 5.65e6T + 4.42e13T^{2}
97 1−6.73e6T+8.07e13T2 1 - 6.73e6T + 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

−9.645844871967169834241509526037, −8.103246128127033904274412980882, −7.42788089894682414656340906366, −6.50315139656960914866137334508, −5.74623681147278097311658343939, −4.73768964142083637401290300871, −3.65824619068420940340202721663, −2.01910948530744845006006600608, −0.62010002493348291489827121907, 0, 0.62010002493348291489827121907, 2.01910948530744845006006600608, 3.65824619068420940340202721663, 4.73768964142083637401290300871, 5.74623681147278097311658343939, 6.50315139656960914866137334508, 7.42788089894682414656340906366, 8.103246128127033904274412980882, 9.645844871967169834241509526037

Graph of the ZZ-function along the critical line