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Properties

Label 2-538-1.1-c7-0-36
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 − 18.4·3-s + 64·4-s + 117.·5-s + 147.·6-s − 512.·7-s − 512·8-s − 1.84e3·9-s − 938.·10-s + 2.50e3·11-s − 1.18e3·12-s + 1.03e4·13-s + 4.09e3·14-s − 2.16e3·15-s + 4.09e3·16-s + 1.23e4·17-s + 1.47e4·18-s + 9.84e3·19-s + 7.50e3·20-s + 9.46e3·21-s − 2.00e4·22-s + 3.43e4·23-s + 9.46e3·24-s − 6.43e4·25-s − 8.28e4·26-s + 7.45e4·27-s − 3.27e4·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.395·3-s + 0.5·4-s + 0.419·5-s + 0.279·6-s − 0.564·7-s − 0.353·8-s − 0.843·9-s − 0.296·10-s + 0.567·11-s − 0.197·12-s + 1.30·13-s + 0.398·14-s − 0.165·15-s + 0.250·16-s + 0.609·17-s + 0.596·18-s + 0.329·19-s + 0.209·20-s + 0.223·21-s − 0.401·22-s + 0.588·23-s + 0.139·24-s − 0.823·25-s − 0.924·26-s + 0.728·27-s − 0.282·28-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 1.3069504291.306950429
L(12)L(\frac12) ≈\approx 1.3069504291.306950429
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+18.4T+2.18e3T2 1 + 18.4T + 2.18e3T^{2}
5 1−117.T+7.81e4T2 1 - 117.T + 7.81e4T^{2}
7 1+512.T+8.23e5T2 1 + 512.T + 8.23e5T^{2}
11 1−2.50e3T+1.94e7T2 1 - 2.50e3T + 1.94e7T^{2}
13 1−1.03e4T+6.27e7T2 1 - 1.03e4T + 6.27e7T^{2}
17 1−1.23e4T+4.10e8T2 1 - 1.23e4T + 4.10e8T^{2}
19 1−9.84e3T+8.93e8T2 1 - 9.84e3T + 8.93e8T^{2}
23 1−3.43e4T+3.40e9T2 1 - 3.43e4T + 3.40e9T^{2}
29 1+1.28e5T+1.72e10T2 1 + 1.28e5T + 1.72e10T^{2}
31 1−1.21e5T+2.75e10T2 1 - 1.21e5T + 2.75e10T^{2}
37 1+2.45e4T+9.49e10T2 1 + 2.45e4T + 9.49e10T^{2}
41 1+4.83e3T+1.94e11T2 1 + 4.83e3T + 1.94e11T^{2}
43 1−2.49e5T+2.71e11T2 1 - 2.49e5T + 2.71e11T^{2}
47 1+6.11e5T+5.06e11T2 1 + 6.11e5T + 5.06e11T^{2}
53 1−1.99e6T+1.17e12T2 1 - 1.99e6T + 1.17e12T^{2}
59 1−6.92e5T+2.48e12T2 1 - 6.92e5T + 2.48e12T^{2}
61 1−1.73e6T+3.14e12T2 1 - 1.73e6T + 3.14e12T^{2}
67 1+4.82e6T+6.06e12T2 1 + 4.82e6T + 6.06e12T^{2}
71 1+1.25e6T+9.09e12T2 1 + 1.25e6T + 9.09e12T^{2}
73 1−3.40e6T+1.10e13T2 1 - 3.40e6T + 1.10e13T^{2}
79 1−2.42e6T+1.92e13T2 1 - 2.42e6T + 1.92e13T^{2}
83 1+6.68e6T+2.71e13T2 1 + 6.68e6T + 2.71e13T^{2}
89 1+1.86e6T+4.42e13T2 1 + 1.86e6T + 4.42e13T^{2}
97 1+9.42e6T+8.07e13T2 1 + 9.42e6T + 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

−9.621264745609679783255093476209, −8.899423135919056531816851321652, −8.082718474129700718089821051223, −6.89249408566860949493993017443, −6.07674949378045955544354078615, −5.48042703033764586019681437766, −3.81700848404806076656094315409, −2.86872877441020267250955950681, −1.53355925452420474651908253292, −0.58937401469545247397041395679, 0.58937401469545247397041395679, 1.53355925452420474651908253292, 2.86872877441020267250955950681, 3.81700848404806076656094315409, 5.48042703033764586019681437766, 6.07674949378045955544354078615, 6.89249408566860949493993017443, 8.082718474129700718089821051223, 8.899423135919056531816851321652, 9.621264745609679783255093476209

Graph of the ZZ-function along the critical line