↑

Properties

Label 2-13-1.1-c9-0-7
Degree 22
Conductor 1313
Sign −1-1
Analytic cond. 6.695466.69546
Root an. cond. 2.587552.58755
Motivic weight 99
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 7.35·2-s + 42.6·3-s − 457.·4-s − 1.23e3·5-s + 313.·6-s + 892.·7-s − 7.13e3·8-s − 1.78e4·9-s − 9.09e3·10-s − 2.71e4·11-s − 1.95e4·12-s − 2.85e4·13-s + 6.56e3·14-s − 5.26e4·15-s + 1.81e5·16-s − 3.46e4·17-s − 1.31e5·18-s + 4.28e5·19-s + 5.66e5·20-s + 3.80e4·21-s − 1.99e5·22-s + 2.03e6·23-s − 3.04e5·24-s − 4.24e5·25-s − 2.10e5·26-s − 1.60e6·27-s − 4.08e5·28-s + ⋯
L(s)  = 1  + 0.325·2-s + 0.303·3-s − 0.894·4-s − 0.884·5-s + 0.0987·6-s + 0.140·7-s − 0.615·8-s − 0.907·9-s − 0.287·10-s − 0.559·11-s − 0.271·12-s − 0.277·13-s + 0.0456·14-s − 0.268·15-s + 0.694·16-s − 0.100·17-s − 0.295·18-s + 0.755·19-s + 0.791·20-s + 0.0426·21-s − 0.181·22-s + 1.51·23-s − 0.187·24-s − 0.217·25-s − 0.0901·26-s − 0.579·27-s − 0.125·28-s + ⋯

Functional equation

Λ(s)=(13s/2 ΓC(s) L(s)=(− Λ(10−s)\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}
Λ(s)=(13s/2 ΓC(s+9/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 1313
Sign: −1-1
Analytic conductor: 6.695466.69546
Root analytic conductor: 2.587552.58755
Motivic weight: 99
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 13, ( :9/2), −1)(2,\ 13,\ (\ :9/2),\ -1)

Particular Values

L(5)L(5) == 00
L(12)L(\frac12) == 00
L(112)L(\frac{11}{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)
bad13 1+2.85e4T 1 + 2.85e4T
good2 1−7.35T+512T2 1 - 7.35T + 512T^{2}
3 1−42.6T+1.96e4T2 1 - 42.6T + 1.96e4T^{2}
5 1+1.23e3T+1.95e6T2 1 + 1.23e3T + 1.95e6T^{2}
7 1−892.T+4.03e7T2 1 - 892.T + 4.03e7T^{2}
11 1+2.71e4T+2.35e9T2 1 + 2.71e4T + 2.35e9T^{2}
17 1+3.46e4T+1.18e11T2 1 + 3.46e4T + 1.18e11T^{2}
19 1−4.28e5T+3.22e11T2 1 - 4.28e5T + 3.22e11T^{2}
23 1−2.03e6T+1.80e12T2 1 - 2.03e6T + 1.80e12T^{2}
29 1+5.26e6T+1.45e13T2 1 + 5.26e6T + 1.45e13T^{2}
31 1+4.15e6T+2.64e13T2 1 + 4.15e6T + 2.64e13T^{2}
37 1+7.58e6T+1.29e14T2 1 + 7.58e6T + 1.29e14T^{2}
41 1+4.92e6T+3.27e14T2 1 + 4.92e6T + 3.27e14T^{2}
43 1−1.71e7T+5.02e14T2 1 - 1.71e7T + 5.02e14T^{2}
47 1+2.95e7T+1.11e15T2 1 + 2.95e7T + 1.11e15T^{2}
53 1+2.72e7T+3.29e15T2 1 + 2.72e7T + 3.29e15T^{2}
59 1+1.13e8T+8.66e15T2 1 + 1.13e8T + 8.66e15T^{2}
61 1+3.76e7T+1.16e16T2 1 + 3.76e7T + 1.16e16T^{2}
67 1−1.90e8T+2.72e16T2 1 - 1.90e8T + 2.72e16T^{2}
71 1−6.87e7T+4.58e16T2 1 - 6.87e7T + 4.58e16T^{2}
73 1−3.61e8T+5.88e16T2 1 - 3.61e8T + 5.88e16T^{2}
79 1+1.42e8T+1.19e17T2 1 + 1.42e8T + 1.19e17T^{2}
83 1+5.80e7T+1.86e17T2 1 + 5.80e7T + 1.86e17T^{2}
89 1+8.59e8T+3.50e17T2 1 + 8.59e8T + 3.50e17T^{2}
97 1−1.46e9T+7.60e17T2 1 - 1.46e9T + 7.60e17T^{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

−17.07393907198354444207628956339, −15.33256819579566625066153488249, −14.22505092003998841012283411943, −12.85445491453793972570809894225, −11.31969655516604555701088514367, −9.196526681358649258536784772497, −7.82905030775083749862309007042, −5.19631160721890757832532363900, −3.38532871414856603989847282361, 0, 3.38532871414856603989847282361, 5.19631160721890757832532363900, 7.82905030775083749862309007042, 9.196526681358649258536784772497, 11.31969655516604555701088514367, 12.85445491453793972570809894225, 14.22505092003998841012283411943, 15.33256819579566625066153488249, 17.07393907198354444207628956339

Graph of the ZZ-function along the critical line