↑

Properties

Label 2-245-1.1-c9-0-27
Degree 22
Conductor 245245
Sign 11
Analytic cond. 126.183126.183
Root an. cond. 11.233111.2331
Motivic weight 99
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2.56·2-s − 177.·3-s − 505.·4-s − 625·5-s + 454.·6-s + 2.60e3·8-s + 1.18e4·9-s + 1.60e3·10-s + 6.14e4·11-s + 8.97e4·12-s + 1.24e5·13-s + 1.10e5·15-s + 2.52e5·16-s + 3.78e5·17-s − 3.03e4·18-s − 7.23e5·19-s + 3.15e5·20-s − 1.57e5·22-s + 5.50e5·23-s − 4.62e5·24-s + 3.90e5·25-s − 3.17e5·26-s + 1.39e6·27-s − 3.62e6·29-s − 2.84e5·30-s − 9.99e6·31-s − 1.98e6·32-s + ⋯
L(s)  = 1  − 0.113·2-s − 1.26·3-s − 0.987·4-s − 0.447·5-s + 0.143·6-s + 0.224·8-s + 0.601·9-s + 0.0506·10-s + 1.26·11-s + 1.24·12-s + 1.20·13-s + 0.565·15-s + 0.961·16-s + 1.10·17-s − 0.0681·18-s − 1.27·19-s + 0.441·20-s − 0.143·22-s + 0.410·23-s − 0.284·24-s + 0.200·25-s − 0.136·26-s + 0.504·27-s − 0.952·29-s − 0.0640·30-s − 1.94·31-s − 0.333·32-s + ⋯

Functional equation

Λ(s)=(245s/2 ΓC(s) L(s)=( Λ(10−s)\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}
Λ(s)=(245s/2 ΓC(s+9/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 245245    =    5⋅725 \cdot 7^{2}
Sign: 11
Analytic conductor: 126.183126.183
Root analytic conductor: 11.233111.2331
Motivic weight: 99
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 245, ( :9/2), 1)(2,\ 245,\ (\ :9/2),\ 1)

Particular Values

L(5)L(5) ≈\approx 0.84105309270.8410530927
L(12)L(\frac12) ≈\approx 0.84105309270.8410530927
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)
bad5 1+625T 1 + 625T
7 1 1
good2 1+2.56T+512T2 1 + 2.56T + 512T^{2}
3 1+177.T+1.96e4T2 1 + 177.T + 1.96e4T^{2}
11 1−6.14e4T+2.35e9T2 1 - 6.14e4T + 2.35e9T^{2}
13 1−1.24e5T+1.06e10T2 1 - 1.24e5T + 1.06e10T^{2}
17 1−3.78e5T+1.18e11T2 1 - 3.78e5T + 1.18e11T^{2}
19 1+7.23e5T+3.22e11T2 1 + 7.23e5T + 3.22e11T^{2}
23 1−5.50e5T+1.80e12T2 1 - 5.50e5T + 1.80e12T^{2}
29 1+3.62e6T+1.45e13T2 1 + 3.62e6T + 1.45e13T^{2}
31 1+9.99e6T+2.64e13T2 1 + 9.99e6T + 2.64e13T^{2}
37 1−4.29e6T+1.29e14T2 1 - 4.29e6T + 1.29e14T^{2}
41 1−3.15e7T+3.27e14T2 1 - 3.15e7T + 3.27e14T^{2}
43 1−2.85e7T+5.02e14T2 1 - 2.85e7T + 5.02e14T^{2}
47 1+2.53e5T+1.11e15T2 1 + 2.53e5T + 1.11e15T^{2}
53 1+3.77e7T+3.29e15T2 1 + 3.77e7T + 3.29e15T^{2}
59 1−5.52e6T+8.66e15T2 1 - 5.52e6T + 8.66e15T^{2}
61 1+1.08e8T+1.16e16T2 1 + 1.08e8T + 1.16e16T^{2}
67 1−1.05e8T+2.72e16T2 1 - 1.05e8T + 2.72e16T^{2}
71 1−3.77e8T+4.58e16T2 1 - 3.77e8T + 4.58e16T^{2}
73 1+3.95e8T+5.88e16T2 1 + 3.95e8T + 5.88e16T^{2}
79 1+1.08e8T+1.19e17T2 1 + 1.08e8T + 1.19e17T^{2}
83 1−1.26e8T+1.86e17T2 1 - 1.26e8T + 1.86e17T^{2}
89 1−2.22e8T+3.50e17T2 1 - 2.22e8T + 3.50e17T^{2}
97 1+1.34e9T+7.60e17T2 1 + 1.34e9T + 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

−10.80206041857004819452617616579, −9.434262635526617204999795394434, −8.714290386933416191665124305034, −7.51879056468725155373922639714, −6.22056948025931053820358057285, −5.56370759611844179226084202684, −4.31438927912138700026873868355, −3.63236121623922028486813591962, −1.35336730623670078019319495982, −0.52896795033934380443309579938, 0.52896795033934380443309579938, 1.35336730623670078019319495982, 3.63236121623922028486813591962, 4.31438927912138700026873868355, 5.56370759611844179226084202684, 6.22056948025931053820358057285, 7.51879056468725155373922639714, 8.714290386933416191665124305034, 9.434262635526617204999795394434, 10.80206041857004819452617616579

Graph of the ZZ-function along the critical line