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Properties

Label 2-538-1.1-c7-0-56
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 + 55.4·3-s + 64·4-s + 70.1·5-s − 443.·6-s + 944.·7-s − 512·8-s + 882.·9-s − 561.·10-s − 4.31e3·11-s + 3.54e3·12-s − 1.84e3·13-s − 7.55e3·14-s + 3.88e3·15-s + 4.09e3·16-s + 2.73e4·17-s − 7.05e3·18-s − 3.19e4·19-s + 4.49e3·20-s + 5.23e4·21-s + 3.44e4·22-s − 1.87e4·23-s − 2.83e4·24-s − 7.31e4·25-s + 1.47e4·26-s − 7.22e4·27-s + 6.04e4·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.18·3-s + 0.5·4-s + 0.251·5-s − 0.837·6-s + 1.04·7-s − 0.353·8-s + 0.403·9-s − 0.177·10-s − 0.976·11-s + 0.592·12-s − 0.232·13-s − 0.736·14-s + 0.297·15-s + 0.250·16-s + 1.34·17-s − 0.285·18-s − 1.06·19-s + 0.125·20-s + 1.23·21-s + 0.690·22-s − 0.321·23-s − 0.418·24-s − 0.936·25-s + 0.164·26-s − 0.706·27-s + 0.520·28-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 2.8531741882.853174188
L(12)L(\frac12) ≈\approx 2.8531741882.853174188
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 1−55.4T+2.18e3T2 1 - 55.4T + 2.18e3T^{2}
5 1−70.1T+7.81e4T2 1 - 70.1T + 7.81e4T^{2}
7 1−944.T+8.23e5T2 1 - 944.T + 8.23e5T^{2}
11 1+4.31e3T+1.94e7T2 1 + 4.31e3T + 1.94e7T^{2}
13 1+1.84e3T+6.27e7T2 1 + 1.84e3T + 6.27e7T^{2}
17 1−2.73e4T+4.10e8T2 1 - 2.73e4T + 4.10e8T^{2}
19 1+3.19e4T+8.93e8T2 1 + 3.19e4T + 8.93e8T^{2}
23 1+1.87e4T+3.40e9T2 1 + 1.87e4T + 3.40e9T^{2}
29 1−8.50e4T+1.72e10T2 1 - 8.50e4T + 1.72e10T^{2}
31 1−8.44e4T+2.75e10T2 1 - 8.44e4T + 2.75e10T^{2}
37 1−6.79e4T+9.49e10T2 1 - 6.79e4T + 9.49e10T^{2}
41 1−4.44e5T+1.94e11T2 1 - 4.44e5T + 1.94e11T^{2}
43 1−9.72e5T+2.71e11T2 1 - 9.72e5T + 2.71e11T^{2}
47 1+1.01e6T+5.06e11T2 1 + 1.01e6T + 5.06e11T^{2}
53 1−1.56e6T+1.17e12T2 1 - 1.56e6T + 1.17e12T^{2}
59 1−1.98e5T+2.48e12T2 1 - 1.98e5T + 2.48e12T^{2}
61 1−9.86e5T+3.14e12T2 1 - 9.86e5T + 3.14e12T^{2}
67 1−1.95e6T+6.06e12T2 1 - 1.95e6T + 6.06e12T^{2}
71 1−8.78e5T+9.09e12T2 1 - 8.78e5T + 9.09e12T^{2}
73 1−3.72e6T+1.10e13T2 1 - 3.72e6T + 1.10e13T^{2}
79 1−4.48e6T+1.92e13T2 1 - 4.48e6T + 1.92e13T^{2}
83 1−8.26e6T+2.71e13T2 1 - 8.26e6T + 2.71e13T^{2}
89 1−1.72e6T+4.42e13T2 1 - 1.72e6T + 4.42e13T^{2}
97 1−2.55e6T+8.07e13T2 1 - 2.55e6T + 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

−9.616869829508804046323109149613, −8.641226063150122774696958290359, −7.924683956290719094740365429675, −7.64884229222007036185068225347, −6.12750281022285702832856943492, −5.07884344599562627498072180903, −3.79076573830019125709081936015, −2.55715062105100101604525192906, −2.02575210920403935476078143955, −0.75866401028503211450234978664, 0.75866401028503211450234978664, 2.02575210920403935476078143955, 2.55715062105100101604525192906, 3.79076573830019125709081936015, 5.07884344599562627498072180903, 6.12750281022285702832856943492, 7.64884229222007036185068225347, 7.924683956290719094740365429675, 8.641226063150122774696958290359, 9.616869829508804046323109149613

Graph of the ZZ-function along the critical line