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Properties

Label 2-538-1.1-c7-0-2
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 − 86.3·3-s + 64·4-s − 4.83·5-s + 691.·6-s − 1.23e3·7-s − 512·8-s + 5.27e3·9-s + 38.6·10-s − 2.29e3·11-s − 5.52e3·12-s − 6.23e3·13-s + 9.87e3·14-s + 417.·15-s + 4.09e3·16-s + 4.48e3·17-s − 4.22e4·18-s − 1.12e4·19-s − 309.·20-s + 1.06e5·21-s + 1.83e4·22-s + 6.00e4·23-s + 4.42e4·24-s − 7.81e4·25-s + 4.98e4·26-s − 2.66e5·27-s − 7.90e4·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.84·3-s + 0.5·4-s − 0.0172·5-s + 1.30·6-s − 1.36·7-s − 0.353·8-s + 2.41·9-s + 0.0122·10-s − 0.520·11-s − 0.923·12-s − 0.786·13-s + 0.961·14-s + 0.0319·15-s + 0.250·16-s + 0.221·17-s − 1.70·18-s − 0.375·19-s − 0.00864·20-s + 2.51·21-s + 0.367·22-s + 1.02·23-s + 0.653·24-s − 0.999·25-s + 0.556·26-s − 2.60·27-s − 0.680·28-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 0.00039308924590.0003930892459
L(12)L(\frac12) ≈\approx 0.00039308924590.0003930892459
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+86.3T+2.18e3T2 1 + 86.3T + 2.18e3T^{2}
5 1+4.83T+7.81e4T2 1 + 4.83T + 7.81e4T^{2}
7 1+1.23e3T+8.23e5T2 1 + 1.23e3T + 8.23e5T^{2}
11 1+2.29e3T+1.94e7T2 1 + 2.29e3T + 1.94e7T^{2}
13 1+6.23e3T+6.27e7T2 1 + 6.23e3T + 6.27e7T^{2}
17 1−4.48e3T+4.10e8T2 1 - 4.48e3T + 4.10e8T^{2}
19 1+1.12e4T+8.93e8T2 1 + 1.12e4T + 8.93e8T^{2}
23 1−6.00e4T+3.40e9T2 1 - 6.00e4T + 3.40e9T^{2}
29 1+1.76e5T+1.72e10T2 1 + 1.76e5T + 1.72e10T^{2}
31 1−2.04e5T+2.75e10T2 1 - 2.04e5T + 2.75e10T^{2}
37 1−2.83e5T+9.49e10T2 1 - 2.83e5T + 9.49e10T^{2}
41 1+7.55e5T+1.94e11T2 1 + 7.55e5T + 1.94e11T^{2}
43 1+1.37e5T+2.71e11T2 1 + 1.37e5T + 2.71e11T^{2}
47 1−1.21e6T+5.06e11T2 1 - 1.21e6T + 5.06e11T^{2}
53 1−3.79e5T+1.17e12T2 1 - 3.79e5T + 1.17e12T^{2}
59 1+3.24e5T+2.48e12T2 1 + 3.24e5T + 2.48e12T^{2}
61 1+1.55e6T+3.14e12T2 1 + 1.55e6T + 3.14e12T^{2}
67 1−1.52e5T+6.06e12T2 1 - 1.52e5T + 6.06e12T^{2}
71 1+5.60e6T+9.09e12T2 1 + 5.60e6T + 9.09e12T^{2}
73 1+5.79e6T+1.10e13T2 1 + 5.79e6T + 1.10e13T^{2}
79 1+6.48e6T+1.92e13T2 1 + 6.48e6T + 1.92e13T^{2}
83 1+4.69e6T+2.71e13T2 1 + 4.69e6T + 2.71e13T^{2}
89 1+1.02e7T+4.42e13T2 1 + 1.02e7T + 4.42e13T^{2}
97 1−5.04e6T+8.07e13T2 1 - 5.04e6T + 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

−10.08314569752242470474043924274, −9.106522143266597436040771665486, −7.56572110094194766778424663484, −6.91679915063996029199196590254, −6.08673457023407909021041057301, −5.41512716139458920547621589343, −4.23273676966403631412037867182, −2.79585078923602657906985556277, −1.32047705280584305721065365342, −0.01132391058577910467166311505, 0.01132391058577910467166311505, 1.32047705280584305721065365342, 2.79585078923602657906985556277, 4.23273676966403631412037867182, 5.41512716139458920547621589343, 6.08673457023407909021041057301, 6.91679915063996029199196590254, 7.56572110094194766778424663484, 9.106522143266597436040771665486, 10.08314569752242470474043924274

Graph of the ZZ-function along the critical line