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Properties

Label 2-538-1.1-c7-0-3
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 − 63.0·3-s + 64·4-s − 242.·5-s + 504.·6-s − 1.04e3·7-s − 512·8-s + 1.78e3·9-s + 1.94e3·10-s − 776.·11-s − 4.03e3·12-s + 4.34e3·13-s + 8.36e3·14-s + 1.52e4·15-s + 4.09e3·16-s − 2.12e4·17-s − 1.42e4·18-s + 4.05e4·19-s − 1.55e4·20-s + 6.59e4·21-s + 6.21e3·22-s + 3.43e4·23-s + 3.22e4·24-s − 1.92e4·25-s − 3.47e4·26-s + 2.52e4·27-s − 6.69e4·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.34·3-s + 0.5·4-s − 0.867·5-s + 0.953·6-s − 1.15·7-s − 0.353·8-s + 0.816·9-s + 0.613·10-s − 0.175·11-s − 0.673·12-s + 0.548·13-s + 0.814·14-s + 1.16·15-s + 0.250·16-s − 1.04·17-s − 0.577·18-s + 1.35·19-s − 0.433·20-s + 1.55·21-s + 0.124·22-s + 0.588·23-s + 0.476·24-s − 0.247·25-s − 0.387·26-s + 0.247·27-s − 0.576·28-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 0.0078017854550.007801785455
L(12)L(\frac12) ≈\approx 0.0078017854550.007801785455
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+63.0T+2.18e3T2 1 + 63.0T + 2.18e3T^{2}
5 1+242.T+7.81e4T2 1 + 242.T + 7.81e4T^{2}
7 1+1.04e3T+8.23e5T2 1 + 1.04e3T + 8.23e5T^{2}
11 1+776.T+1.94e7T2 1 + 776.T + 1.94e7T^{2}
13 1−4.34e3T+6.27e7T2 1 - 4.34e3T + 6.27e7T^{2}
17 1+2.12e4T+4.10e8T2 1 + 2.12e4T + 4.10e8T^{2}
19 1−4.05e4T+8.93e8T2 1 - 4.05e4T + 8.93e8T^{2}
23 1−3.43e4T+3.40e9T2 1 - 3.43e4T + 3.40e9T^{2}
29 1+1.56e5T+1.72e10T2 1 + 1.56e5T + 1.72e10T^{2}
31 1+2.22e5T+2.75e10T2 1 + 2.22e5T + 2.75e10T^{2}
37 1+2.66e4T+9.49e10T2 1 + 2.66e4T + 9.49e10T^{2}
41 1−6.11e5T+1.94e11T2 1 - 6.11e5T + 1.94e11T^{2}
43 1+3.03e5T+2.71e11T2 1 + 3.03e5T + 2.71e11T^{2}
47 1+3.23e5T+5.06e11T2 1 + 3.23e5T + 5.06e11T^{2}
53 1−4.72e5T+1.17e12T2 1 - 4.72e5T + 1.17e12T^{2}
59 1+2.54e6T+2.48e12T2 1 + 2.54e6T + 2.48e12T^{2}
61 1+1.94e6T+3.14e12T2 1 + 1.94e6T + 3.14e12T^{2}
67 1+1.62e6T+6.06e12T2 1 + 1.62e6T + 6.06e12T^{2}
71 1+7.16e5T+9.09e12T2 1 + 7.16e5T + 9.09e12T^{2}
73 1−2.95e6T+1.10e13T2 1 - 2.95e6T + 1.10e13T^{2}
79 1+3.99e6T+1.92e13T2 1 + 3.99e6T + 1.92e13T^{2}
83 1+2.79e6T+2.71e13T2 1 + 2.79e6T + 2.71e13T^{2}
89 1−4.35e6T+4.42e13T2 1 - 4.35e6T + 4.42e13T^{2}
97 1+1.25e7T+8.07e13T2 1 + 1.25e7T + 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

−9.660753254521800594431444760884, −9.010366219169969683029812321146, −7.71364613676056591524906342012, −6.98722602902404824026102473183, −6.15394721524736625554166896495, −5.33681529668123501553140755655, −4.00505048076096530370152087348, −2.99047169705939724383023212430, −1.31952676956878700277165460254, −0.04945787300275544121812853989, 0.04945787300275544121812853989, 1.31952676956878700277165460254, 2.99047169705939724383023212430, 4.00505048076096530370152087348, 5.33681529668123501553140755655, 6.15394721524736625554166896495, 6.98722602902404824026102473183, 7.71364613676056591524906342012, 9.010366219169969683029812321146, 9.660753254521800594431444760884

Graph of the ZZ-function along the critical line