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Properties

Label 2-538-1.1-c7-0-4
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 + 14.3·3-s + 64·4-s + 230.·5-s − 115.·6-s − 1.45e3·7-s − 512·8-s − 1.98e3·9-s − 1.84e3·10-s − 6.48e3·11-s + 920.·12-s − 1.14e3·13-s + 1.16e4·14-s + 3.31e3·15-s + 4.09e3·16-s − 1.01e4·17-s + 1.58e4·18-s − 3.16e4·19-s + 1.47e4·20-s − 2.09e4·21-s + 5.18e4·22-s − 2.13e4·23-s − 7.36e3·24-s − 2.48e4·25-s + 9.13e3·26-s − 5.99e4·27-s − 9.33e4·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.307·3-s + 0.5·4-s + 0.825·5-s − 0.217·6-s − 1.60·7-s − 0.353·8-s − 0.905·9-s − 0.583·10-s − 1.46·11-s + 0.153·12-s − 0.144·13-s + 1.13·14-s + 0.253·15-s + 0.250·16-s − 0.498·17-s + 0.640·18-s − 1.05·19-s + 0.412·20-s − 0.494·21-s + 1.03·22-s − 0.365·23-s − 0.108·24-s − 0.318·25-s + 0.101·26-s − 0.585·27-s − 0.803·28-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 0.044157559000.04415755900
L(12)L(\frac12) ≈\approx 0.044157559000.04415755900
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 1−14.3T+2.18e3T2 1 - 14.3T + 2.18e3T^{2}
5 1−230.T+7.81e4T2 1 - 230.T + 7.81e4T^{2}
7 1+1.45e3T+8.23e5T2 1 + 1.45e3T + 8.23e5T^{2}
11 1+6.48e3T+1.94e7T2 1 + 6.48e3T + 1.94e7T^{2}
13 1+1.14e3T+6.27e7T2 1 + 1.14e3T + 6.27e7T^{2}
17 1+1.01e4T+4.10e8T2 1 + 1.01e4T + 4.10e8T^{2}
19 1+3.16e4T+8.93e8T2 1 + 3.16e4T + 8.93e8T^{2}
23 1+2.13e4T+3.40e9T2 1 + 2.13e4T + 3.40e9T^{2}
29 1+1.89e5T+1.72e10T2 1 + 1.89e5T + 1.72e10T^{2}
31 1+1.55e5T+2.75e10T2 1 + 1.55e5T + 2.75e10T^{2}
37 1+4.02e5T+9.49e10T2 1 + 4.02e5T + 9.49e10T^{2}
41 1−1.20e5T+1.94e11T2 1 - 1.20e5T + 1.94e11T^{2}
43 1−6.31e5T+2.71e11T2 1 - 6.31e5T + 2.71e11T^{2}
47 1−2.21e5T+5.06e11T2 1 - 2.21e5T + 5.06e11T^{2}
53 1+6.77e5T+1.17e12T2 1 + 6.77e5T + 1.17e12T^{2}
59 1+7.34e4T+2.48e12T2 1 + 7.34e4T + 2.48e12T^{2}
61 1−1.56e6T+3.14e12T2 1 - 1.56e6T + 3.14e12T^{2}
67 1+3.28e6T+6.06e12T2 1 + 3.28e6T + 6.06e12T^{2}
71 1−4.47e6T+9.09e12T2 1 - 4.47e6T + 9.09e12T^{2}
73 1−2.65e5T+1.10e13T2 1 - 2.65e5T + 1.10e13T^{2}
79 1+4.53e6T+1.92e13T2 1 + 4.53e6T + 1.92e13T^{2}
83 1−5.34e6T+2.71e13T2 1 - 5.34e6T + 2.71e13T^{2}
89 1+1.44e6T+4.42e13T2 1 + 1.44e6T + 4.42e13T^{2}
97 1−2.72e5T+8.07e13T2 1 - 2.72e5T + 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

−9.532346055413128932297722255061, −9.038444926715759899525593869464, −8.038476086671851981400021966097, −7.04255712267325465498999655445, −6.04415644493371230907431145810, −5.49251350833277353089093645947, −3.67783107496985387406487030819, −2.63505983142150258523696291238, −2.06572409139643410995172418641, −0.089845587069360355831667030783, 0.089845587069360355831667030783, 2.06572409139643410995172418641, 2.63505983142150258523696291238, 3.67783107496985387406487030819, 5.49251350833277353089093645947, 6.04415644493371230907431145810, 7.04255712267325465498999655445, 8.038476086671851981400021966097, 9.038444926715759899525593869464, 9.532346055413128932297722255061

Graph of the ZZ-function along the critical line