↑

Properties

Label 2-538-1.1-c7-0-0
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

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 8·2-s − 25.6·3-s + 64·4-s − 255.·5-s + 204.·6-s − 1.14e3·7-s − 512·8-s − 1.53e3·9-s + 2.04e3·10-s + 1.07e3·11-s − 1.63e3·12-s + 6.57e3·13-s + 9.17e3·14-s + 6.55e3·15-s + 4.09e3·16-s + 1.55e4·17-s + 1.22e4·18-s − 8.17e3·19-s − 1.63e4·20-s + 2.93e4·21-s − 8.60e3·22-s − 2.57e4·23-s + 1.31e4·24-s − 1.26e4·25-s − 5.25e4·26-s + 9.52e4·27-s − 7.34e4·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.547·3-s + 0.5·4-s − 0.915·5-s + 0.387·6-s − 1.26·7-s − 0.353·8-s − 0.700·9-s + 0.647·10-s + 0.243·11-s − 0.273·12-s + 0.829·13-s + 0.893·14-s + 0.501·15-s + 0.250·16-s + 0.767·17-s + 0.495·18-s − 0.273·19-s − 0.457·20-s + 0.692·21-s − 0.172·22-s − 0.441·23-s + 0.193·24-s − 0.162·25-s − 0.586·26-s + 0.931·27-s − 0.631·28-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 6.420329689×10−56.420329689\times10^{-5}
L(12)L(\frac12) ≈\approx 6.420329689×10−56.420329689\times10^{-5}
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+25.6T+2.18e3T2 1 + 25.6T + 2.18e3T^{2}
5 1+255.T+7.81e4T2 1 + 255.T + 7.81e4T^{2}
7 1+1.14e3T+8.23e5T2 1 + 1.14e3T + 8.23e5T^{2}
11 1−1.07e3T+1.94e7T2 1 - 1.07e3T + 1.94e7T^{2}
13 1−6.57e3T+6.27e7T2 1 - 6.57e3T + 6.27e7T^{2}
17 1−1.55e4T+4.10e8T2 1 - 1.55e4T + 4.10e8T^{2}
19 1+8.17e3T+8.93e8T2 1 + 8.17e3T + 8.93e8T^{2}
23 1+2.57e4T+3.40e9T2 1 + 2.57e4T + 3.40e9T^{2}
29 1−2.79e4T+1.72e10T2 1 - 2.79e4T + 1.72e10T^{2}
31 1+1.11e5T+2.75e10T2 1 + 1.11e5T + 2.75e10T^{2}
37 1+5.25e5T+9.49e10T2 1 + 5.25e5T + 9.49e10T^{2}
41 1+6.58e5T+1.94e11T2 1 + 6.58e5T + 1.94e11T^{2}
43 1+7.51e5T+2.71e11T2 1 + 7.51e5T + 2.71e11T^{2}
47 1+6.74e5T+5.06e11T2 1 + 6.74e5T + 5.06e11T^{2}
53 1+1.60e6T+1.17e12T2 1 + 1.60e6T + 1.17e12T^{2}
59 1+1.19e5T+2.48e12T2 1 + 1.19e5T + 2.48e12T^{2}
61 1−1.44e6T+3.14e12T2 1 - 1.44e6T + 3.14e12T^{2}
67 1−1.18e6T+6.06e12T2 1 - 1.18e6T + 6.06e12T^{2}
71 1+4.29e6T+9.09e12T2 1 + 4.29e6T + 9.09e12T^{2}
73 1−1.46e6T+1.10e13T2 1 - 1.46e6T + 1.10e13T^{2}
79 1−6.48e5T+1.92e13T2 1 - 6.48e5T + 1.92e13T^{2}
83 1+5.40e6T+2.71e13T2 1 + 5.40e6T + 2.71e13T^{2}
89 1+8.12e6T+4.42e13T2 1 + 8.12e6T + 4.42e13T^{2}
97 1−4.92e6T+8.07e13T2 1 - 4.92e6T + 8.07e13T^{2}
show more
show less
   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

−9.757754411465696687163485004965, −8.696467095679658309425520004290, −8.091803831270054056331276392497, −6.88882649069295858972879110364, −6.29183758803354606602886032715, −5.28806825267250608762137268700, −3.69193090613411410695224405399, −3.14279692738715134810961129369, −1.49433938210078185515315657726, −0.00403461804112644685241746867, 0.00403461804112644685241746867, 1.49433938210078185515315657726, 3.14279692738715134810961129369, 3.69193090613411410695224405399, 5.28806825267250608762137268700, 6.29183758803354606602886032715, 6.88882649069295858972879110364, 8.091803831270054056331276392497, 8.696467095679658309425520004290, 9.757754411465696687163485004965

Graph of the ZZ-function along the critical line