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Properties

Label 2-538-1.1-c7-0-98
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 + 64.7·3-s + 64·4-s − 436.·5-s − 517.·6-s − 168.·7-s − 512·8-s + 2.00e3·9-s + 3.48e3·10-s − 5.90e3·11-s + 4.14e3·12-s + 4.98e3·13-s + 1.35e3·14-s − 2.82e4·15-s + 4.09e3·16-s + 1.38e4·17-s − 1.60e4·18-s + 3.54e4·19-s − 2.79e4·20-s − 1.09e4·21-s + 4.72e4·22-s + 794.·23-s − 3.31e4·24-s + 1.12e5·25-s − 3.98e4·26-s − 1.20e4·27-s − 1.08e4·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.38·3-s + 0.5·4-s − 1.56·5-s − 0.978·6-s − 0.186·7-s − 0.353·8-s + 0.915·9-s + 1.10·10-s − 1.33·11-s + 0.691·12-s + 0.628·13-s + 0.131·14-s − 2.15·15-s + 0.250·16-s + 0.685·17-s − 0.647·18-s + 1.18·19-s − 0.780·20-s − 0.257·21-s + 0.946·22-s + 0.0136·23-s − 0.489·24-s + 1.43·25-s − 0.444·26-s − 0.117·27-s − 0.0930·28-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 1−64.7T+2.18e3T2 1 - 64.7T + 2.18e3T^{2}
5 1+436.T+7.81e4T2 1 + 436.T + 7.81e4T^{2}
7 1+168.T+8.23e5T2 1 + 168.T + 8.23e5T^{2}
11 1+5.90e3T+1.94e7T2 1 + 5.90e3T + 1.94e7T^{2}
13 1−4.98e3T+6.27e7T2 1 - 4.98e3T + 6.27e7T^{2}
17 1−1.38e4T+4.10e8T2 1 - 1.38e4T + 4.10e8T^{2}
19 1−3.54e4T+8.93e8T2 1 - 3.54e4T + 8.93e8T^{2}
23 1−794.T+3.40e9T2 1 - 794.T + 3.40e9T^{2}
29 1+1.34e5T+1.72e10T2 1 + 1.34e5T + 1.72e10T^{2}
31 1−2.83e5T+2.75e10T2 1 - 2.83e5T + 2.75e10T^{2}
37 1+2.90e5T+9.49e10T2 1 + 2.90e5T + 9.49e10T^{2}
41 1−2.96e5T+1.94e11T2 1 - 2.96e5T + 1.94e11T^{2}
43 1−5.52e5T+2.71e11T2 1 - 5.52e5T + 2.71e11T^{2}
47 1−1.24e6T+5.06e11T2 1 - 1.24e6T + 5.06e11T^{2}
53 1−3.51e4T+1.17e12T2 1 - 3.51e4T + 1.17e12T^{2}
59 1+7.19e5T+2.48e12T2 1 + 7.19e5T + 2.48e12T^{2}
61 1−4.50e5T+3.14e12T2 1 - 4.50e5T + 3.14e12T^{2}
67 1−2.63e6T+6.06e12T2 1 - 2.63e6T + 6.06e12T^{2}
71 1+3.89e6T+9.09e12T2 1 + 3.89e6T + 9.09e12T^{2}
73 1+1.66e6T+1.10e13T2 1 + 1.66e6T + 1.10e13T^{2}
79 1+1.95e6T+1.92e13T2 1 + 1.95e6T + 1.92e13T^{2}
83 1+5.27e6T+2.71e13T2 1 + 5.27e6T + 2.71e13T^{2}
89 1+6.84e6T+4.42e13T2 1 + 6.84e6T + 4.42e13T^{2}
97 1+1.07e6T+8.07e13T2 1 + 1.07e6T + 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

−9.018626829908871255952064118805, −8.208400702806391412426210870411, −7.76436441323046331639059024888, −7.19697706434239061326632652292, −5.57765320761655949895918387344, −4.12968609341996431393932687645, −3.24089932177548019645144658528, −2.63911284277382821389997664787, −1.11972073492575605683185063231, 0, 1.11972073492575605683185063231, 2.63911284277382821389997664787, 3.24089932177548019645144658528, 4.12968609341996431393932687645, 5.57765320761655949895918387344, 7.19697706434239061326632652292, 7.76436441323046331639059024888, 8.208400702806391412426210870411, 9.018626829908871255952064118805

Graph of the ZZ-function along the critical line