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Properties

Label 2-538-1.1-c7-0-104
Degree 22
Conductor 538538
Sign −1-1
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 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·2-s + 44.2·3-s + 64·4-s − 277.·5-s − 354.·6-s + 438.·7-s − 512·8-s − 227.·9-s + 2.21e3·10-s + 3.10e3·11-s + 2.83e3·12-s − 1.27e3·13-s − 3.50e3·14-s − 1.22e4·15-s + 4.09e3·16-s − 1.91e3·17-s + 1.82e3·18-s + 3.40e4·19-s − 1.77e4·20-s + 1.93e4·21-s − 2.48e4·22-s − 4.54e4·23-s − 2.26e4·24-s − 1.16e3·25-s + 1.02e4·26-s − 1.06e5·27-s + 2.80e4·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.946·3-s + 0.5·4-s − 0.992·5-s − 0.669·6-s + 0.482·7-s − 0.353·8-s − 0.104·9-s + 0.701·10-s + 0.703·11-s + 0.473·12-s − 0.161·13-s − 0.341·14-s − 0.939·15-s + 0.250·16-s − 0.0945·17-s + 0.0737·18-s + 1.13·19-s − 0.496·20-s + 0.457·21-s − 0.497·22-s − 0.779·23-s − 0.334·24-s − 0.0148·25-s + 0.114·26-s − 1.04·27-s + 0.241·28-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: −1-1
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: 11
Selberg data: (2, 538, ( :7/2), −1)(2,\ 538,\ (\ :7/2),\ -1)

Particular Values

L(4)L(4) == 00
L(12)L(\frac12) == 00
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 1−44.2T+2.18e3T2 1 - 44.2T + 2.18e3T^{2}
5 1+277.T+7.81e4T2 1 + 277.T + 7.81e4T^{2}
7 1−438.T+8.23e5T2 1 - 438.T + 8.23e5T^{2}
11 1−3.10e3T+1.94e7T2 1 - 3.10e3T + 1.94e7T^{2}
13 1+1.27e3T+6.27e7T2 1 + 1.27e3T + 6.27e7T^{2}
17 1+1.91e3T+4.10e8T2 1 + 1.91e3T + 4.10e8T^{2}
19 1−3.40e4T+8.93e8T2 1 - 3.40e4T + 8.93e8T^{2}
23 1+4.54e4T+3.40e9T2 1 + 4.54e4T + 3.40e9T^{2}
29 1−2.91e4T+1.72e10T2 1 - 2.91e4T + 1.72e10T^{2}
31 1+1.13e5T+2.75e10T2 1 + 1.13e5T + 2.75e10T^{2}
37 1−2.97e5T+9.49e10T2 1 - 2.97e5T + 9.49e10T^{2}
41 1−6.39e5T+1.94e11T2 1 - 6.39e5T + 1.94e11T^{2}
43 1−2.22e4T+2.71e11T2 1 - 2.22e4T + 2.71e11T^{2}
47 1+6.50e5T+5.06e11T2 1 + 6.50e5T + 5.06e11T^{2}
53 1+1.51e5T+1.17e12T2 1 + 1.51e5T + 1.17e12T^{2}
59 1−1.10e6T+2.48e12T2 1 - 1.10e6T + 2.48e12T^{2}
61 1−1.92e6T+3.14e12T2 1 - 1.92e6T + 3.14e12T^{2}
67 1+1.01e6T+6.06e12T2 1 + 1.01e6T + 6.06e12T^{2}
71 1+6.17e4T+9.09e12T2 1 + 6.17e4T + 9.09e12T^{2}
73 1−4.16e6T+1.10e13T2 1 - 4.16e6T + 1.10e13T^{2}
79 1+1.00e6T+1.92e13T2 1 + 1.00e6T + 1.92e13T^{2}
83 1+5.77e6T+2.71e13T2 1 + 5.77e6T + 2.71e13T^{2}
89 1−3.75e6T+4.42e13T2 1 - 3.75e6T + 4.42e13T^{2}
97 1+4.23e6T+8.07e13T2 1 + 4.23e6T + 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

−9.208117447181804666743869588875, −8.145220958667643432028977493186, −7.895744421549559320756928366715, −6.92861465515099769666629499467, −5.64393975560063642951653488979, −4.22318011427521769599177672754, −3.39602495335568132688581614337, −2.35125130267680184970036030328, −1.20160492267832383028633106609, 0, 1.20160492267832383028633106609, 2.35125130267680184970036030328, 3.39602495335568132688581614337, 4.22318011427521769599177672754, 5.64393975560063642951653488979, 6.92861465515099769666629499467, 7.895744421549559320756928366715, 8.145220958667643432028977493186, 9.208117447181804666743869588875

Graph of the ZZ-function along the critical line