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Properties

Label 2-538-1.1-c7-0-61
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 − 47.1·3-s + 64·4-s + 395.·5-s − 377.·6-s + 1.40e3·7-s + 512·8-s + 37.6·9-s + 3.16e3·10-s − 3.67e3·11-s − 3.01e3·12-s − 1.22e4·13-s + 1.12e4·14-s − 1.86e4·15-s + 4.09e3·16-s − 2.35e4·17-s + 300.·18-s + 3.95e4·19-s + 2.52e4·20-s − 6.62e4·21-s − 2.94e4·22-s + 2.64e3·23-s − 2.41e4·24-s + 7.79e4·25-s − 9.83e4·26-s + 1.01e5·27-s + 8.99e4·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.00·3-s + 0.5·4-s + 1.41·5-s − 0.713·6-s + 1.54·7-s + 0.353·8-s + 0.0171·9-s + 0.999·10-s − 0.832·11-s − 0.504·12-s − 1.55·13-s + 1.09·14-s − 1.42·15-s + 0.250·16-s − 1.16·17-s + 0.0121·18-s + 1.32·19-s + 0.706·20-s − 1.56·21-s − 0.588·22-s + 0.0453·23-s − 0.356·24-s + 0.997·25-s − 1.09·26-s + 0.991·27-s + 0.773·28-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 3.6744711253.674471125
L(12)L(\frac12) ≈\approx 3.6744711253.674471125
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+47.1T+2.18e3T2 1 + 47.1T + 2.18e3T^{2}
5 1−395.T+7.81e4T2 1 - 395.T + 7.81e4T^{2}
7 1−1.40e3T+8.23e5T2 1 - 1.40e3T + 8.23e5T^{2}
11 1+3.67e3T+1.94e7T2 1 + 3.67e3T + 1.94e7T^{2}
13 1+1.22e4T+6.27e7T2 1 + 1.22e4T + 6.27e7T^{2}
17 1+2.35e4T+4.10e8T2 1 + 2.35e4T + 4.10e8T^{2}
19 1−3.95e4T+8.93e8T2 1 - 3.95e4T + 8.93e8T^{2}
23 1−2.64e3T+3.40e9T2 1 - 2.64e3T + 3.40e9T^{2}
29 1−3.18e4T+1.72e10T2 1 - 3.18e4T + 1.72e10T^{2}
31 1−2.85e5T+2.75e10T2 1 - 2.85e5T + 2.75e10T^{2}
37 1+5.04e5T+9.49e10T2 1 + 5.04e5T + 9.49e10T^{2}
41 1+8.75e4T+1.94e11T2 1 + 8.75e4T + 1.94e11T^{2}
43 1−4.74e5T+2.71e11T2 1 - 4.74e5T + 2.71e11T^{2}
47 1−6.80e5T+5.06e11T2 1 - 6.80e5T + 5.06e11T^{2}
53 1+1.09e5T+1.17e12T2 1 + 1.09e5T + 1.17e12T^{2}
59 1−1.50e6T+2.48e12T2 1 - 1.50e6T + 2.48e12T^{2}
61 1−2.38e6T+3.14e12T2 1 - 2.38e6T + 3.14e12T^{2}
67 1−2.45e6T+6.06e12T2 1 - 2.45e6T + 6.06e12T^{2}
71 1+4.15e6T+9.09e12T2 1 + 4.15e6T + 9.09e12T^{2}
73 1+5.34e6T+1.10e13T2 1 + 5.34e6T + 1.10e13T^{2}
79 1−3.64e6T+1.92e13T2 1 - 3.64e6T + 1.92e13T^{2}
83 1−6.93e6T+2.71e13T2 1 - 6.93e6T + 2.71e13T^{2}
89 1−6.11e6T+4.42e13T2 1 - 6.11e6T + 4.42e13T^{2}
97 1−1.79e7T+8.07e13T2 1 - 1.79e7T + 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

−10.10400753599642899101702765634, −8.830533754242792770816567408938, −7.65229877099166084907076503783, −6.72762857160128056577883988915, −5.66197897792254238854864223435, −5.10698494641493229434436499239, −4.69190444998912304757557745145, −2.64627666828929645758140809602, −2.01169691526399696709088351941, −0.78192437749901104675321600759, 0.78192437749901104675321600759, 2.01169691526399696709088351941, 2.64627666828929645758140809602, 4.69190444998912304757557745145, 5.10698494641493229434436499239, 5.66197897792254238854864223435, 6.72762857160128056577883988915, 7.65229877099166084907076503783, 8.830533754242792770816567408938, 10.10400753599642899101702765634

Graph of the ZZ-function along the critical line