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Properties

Label 2-538-1.1-c7-0-14
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 + 13.5·3-s + 64·4-s − 206.·5-s + 108.·6-s − 1.55e3·7-s + 512·8-s − 2.00e3·9-s − 1.65e3·10-s − 826.·11-s + 866.·12-s − 1.27e4·13-s − 1.24e4·14-s − 2.80e3·15-s + 4.09e3·16-s + 3.25e4·17-s − 1.60e4·18-s − 5.80e4·19-s − 1.32e4·20-s − 2.10e4·21-s − 6.61e3·22-s − 108.·23-s + 6.93e3·24-s − 3.53e4·25-s − 1.01e5·26-s − 5.67e4·27-s − 9.92e4·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.289·3-s + 0.5·4-s − 0.740·5-s + 0.204·6-s − 1.70·7-s + 0.353·8-s − 0.916·9-s − 0.523·10-s − 0.187·11-s + 0.144·12-s − 1.60·13-s − 1.20·14-s − 0.214·15-s + 0.250·16-s + 1.60·17-s − 0.647·18-s − 1.94·19-s − 0.370·20-s − 0.494·21-s − 0.132·22-s − 0.00186·23-s + 0.102·24-s − 0.452·25-s − 1.13·26-s − 0.554·27-s − 0.854·28-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 0.70337163330.7033716333
L(12)L(\frac12) ≈\approx 0.70337163330.7033716333
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 1−13.5T+2.18e3T2 1 - 13.5T + 2.18e3T^{2}
5 1+206.T+7.81e4T2 1 + 206.T + 7.81e4T^{2}
7 1+1.55e3T+8.23e5T2 1 + 1.55e3T + 8.23e5T^{2}
11 1+826.T+1.94e7T2 1 + 826.T + 1.94e7T^{2}
13 1+1.27e4T+6.27e7T2 1 + 1.27e4T + 6.27e7T^{2}
17 1−3.25e4T+4.10e8T2 1 - 3.25e4T + 4.10e8T^{2}
19 1+5.80e4T+8.93e8T2 1 + 5.80e4T + 8.93e8T^{2}
23 1+108.T+3.40e9T2 1 + 108.T + 3.40e9T^{2}
29 1+5.48e4T+1.72e10T2 1 + 5.48e4T + 1.72e10T^{2}
31 1−3.22e5T+2.75e10T2 1 - 3.22e5T + 2.75e10T^{2}
37 1+4.39e5T+9.49e10T2 1 + 4.39e5T + 9.49e10T^{2}
41 1+7.58e5T+1.94e11T2 1 + 7.58e5T + 1.94e11T^{2}
43 1−7.92e4T+2.71e11T2 1 - 7.92e4T + 2.71e11T^{2}
47 1+2.34e5T+5.06e11T2 1 + 2.34e5T + 5.06e11T^{2}
53 1−5.78e5T+1.17e12T2 1 - 5.78e5T + 1.17e12T^{2}
59 1−2.54e6T+2.48e12T2 1 - 2.54e6T + 2.48e12T^{2}
61 1+6.68e5T+3.14e12T2 1 + 6.68e5T + 3.14e12T^{2}
67 1+2.35e6T+6.06e12T2 1 + 2.35e6T + 6.06e12T^{2}
71 1+7.49e5T+9.09e12T2 1 + 7.49e5T + 9.09e12T^{2}
73 1+1.40e6T+1.10e13T2 1 + 1.40e6T + 1.10e13T^{2}
79 1−3.21e6T+1.92e13T2 1 - 3.21e6T + 1.92e13T^{2}
83 1−1.97e5T+2.71e13T2 1 - 1.97e5T + 2.71e13T^{2}
89 1−6.87e6T+4.42e13T2 1 - 6.87e6T + 4.42e13T^{2}
97 1−7.02e6T+8.07e13T2 1 - 7.02e6T + 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

−9.937248080429228287156939567499, −8.704636934418600274310861084284, −7.80283273108565267581732450180, −6.86482220922954964779493722181, −6.05201570148342279713416389915, −5.02459763050559563540063172551, −3.80501122964181138781046899621, −3.11382718845917595773140992636, −2.31062949533681018119714304327, −0.29301971281268955295600507413, 0.29301971281268955295600507413, 2.31062949533681018119714304327, 3.11382718845917595773140992636, 3.80501122964181138781046899621, 5.02459763050559563540063172551, 6.05201570148342279713416389915, 6.86482220922954964779493722181, 7.80283273108565267581732450180, 8.704636934418600274310861084284, 9.937248080429228287156939567499

Graph of the ZZ-function along the critical line