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Properties

Label 2-538-1.1-c5-0-3
Degree 22
Conductor 538538
Sign 11
Analytic cond. 86.286486.2864
Root an. cond. 9.289059.28905
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s + 19.2·3-s + 16·4-s − 81.3·5-s − 76.8·6-s − 137.·7-s − 64·8-s + 126.·9-s + 325.·10-s − 458.·11-s + 307.·12-s − 1.04e3·13-s + 548.·14-s − 1.56e3·15-s + 256·16-s − 1.14e3·17-s − 505.·18-s + 1.26e3·19-s − 1.30e3·20-s − 2.63e3·21-s + 1.83e3·22-s + 2.58e3·23-s − 1.22e3·24-s + 3.49e3·25-s + 4.18e3·26-s − 2.24e3·27-s − 2.19e3·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.23·3-s + 0.5·4-s − 1.45·5-s − 0.871·6-s − 1.05·7-s − 0.353·8-s + 0.519·9-s + 1.02·10-s − 1.14·11-s + 0.616·12-s − 1.71·13-s + 0.747·14-s − 1.79·15-s + 0.250·16-s − 0.963·17-s − 0.367·18-s + 0.801·19-s − 0.727·20-s − 1.30·21-s + 0.807·22-s + 1.01·23-s − 0.435·24-s + 1.11·25-s + 1.21·26-s − 0.591·27-s − 0.528·28-s + ⋯

Functional equation

Λ(s)=(538s/2 ΓC(s) L(s)=( Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(538s/2 ΓC(s+5/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 538538    =    2⋅2692 \cdot 269
Sign: 11
Analytic conductor: 86.286486.2864
Root analytic conductor: 9.289059.28905
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 538, ( :5/2), 1)(2,\ 538,\ (\ :5/2),\ 1)

Particular Values

L(3)L(3) ≈\approx 0.32840511140.3284051114
L(12)L(\frac12) ≈\approx 0.32840511140.3284051114
L(72)L(\frac{7}{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+4T 1 + 4T
269 1−7.23e4T 1 - 7.23e4T
good3 1−19.2T+243T2 1 - 19.2T + 243T^{2}
5 1+81.3T+3.12e3T2 1 + 81.3T + 3.12e3T^{2}
7 1+137.T+1.68e4T2 1 + 137.T + 1.68e4T^{2}
11 1+458.T+1.61e5T2 1 + 458.T + 1.61e5T^{2}
13 1+1.04e3T+3.71e5T2 1 + 1.04e3T + 3.71e5T^{2}
17 1+1.14e3T+1.41e6T2 1 + 1.14e3T + 1.41e6T^{2}
19 1−1.26e3T+2.47e6T2 1 - 1.26e3T + 2.47e6T^{2}
23 1−2.58e3T+6.43e6T2 1 - 2.58e3T + 6.43e6T^{2}
29 1+5.99e3T+2.05e7T2 1 + 5.99e3T + 2.05e7T^{2}
31 1+1.03e4T+2.86e7T2 1 + 1.03e4T + 2.86e7T^{2}
37 1−7.87e3T+6.93e7T2 1 - 7.87e3T + 6.93e7T^{2}
41 1−1.18e4T+1.15e8T2 1 - 1.18e4T + 1.15e8T^{2}
43 1−1.77e3T+1.47e8T2 1 - 1.77e3T + 1.47e8T^{2}
47 1+1.99e3T+2.29e8T2 1 + 1.99e3T + 2.29e8T^{2}
53 1+2.44e3T+4.18e8T2 1 + 2.44e3T + 4.18e8T^{2}
59 1−3.67e4T+7.14e8T2 1 - 3.67e4T + 7.14e8T^{2}
61 1+5.77e3T+8.44e8T2 1 + 5.77e3T + 8.44e8T^{2}
67 1−3.77e4T+1.35e9T2 1 - 3.77e4T + 1.35e9T^{2}
71 1−6.55e4T+1.80e9T2 1 - 6.55e4T + 1.80e9T^{2}
73 1+4.05e4T+2.07e9T2 1 + 4.05e4T + 2.07e9T^{2}
79 1+5.72e4T+3.07e9T2 1 + 5.72e4T + 3.07e9T^{2}
83 1+1.00e5T+3.93e9T2 1 + 1.00e5T + 3.93e9T^{2}
89 1−1.00e5T+5.58e9T2 1 - 1.00e5T + 5.58e9T^{2}
97 1−1.80e4T+8.58e9T2 1 - 1.80e4T + 8.58e9T^{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

−9.599191595366403425585757367439, −9.253342247224145437157107225909, −8.184289053608584052520068057962, −7.44077416596396879740088370490, −7.13355113049131521861183736048, −5.33636859563493964248740655653, −3.94967358938449034808501687080, −3.01585912728635301486886888670, −2.35549626940297673692731645523, −0.27214252504719193454276988243, 0.27214252504719193454276988243, 2.35549626940297673692731645523, 3.01585912728635301486886888670, 3.94967358938449034808501687080, 5.33636859563493964248740655653, 7.13355113049131521861183736048, 7.44077416596396879740088370490, 8.184289053608584052520068057962, 9.253342247224145437157107225909, 9.599191595366403425585757367439

Graph of the ZZ-function along the critical line