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Properties

Label 2-245-1.1-c9-0-19
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 34.1·2-s − 128.·3-s + 652.·4-s + 625·5-s + 4.38e3·6-s − 4.80e3·8-s − 3.21e3·9-s − 2.13e4·10-s − 2.66e4·11-s − 8.37e4·12-s + 1.32e5·13-s − 8.02e4·15-s − 1.70e5·16-s − 6.18e5·17-s + 1.09e5·18-s + 8.29e5·19-s + 4.07e5·20-s + 9.11e5·22-s − 1.47e6·23-s + 6.16e5·24-s + 3.90e5·25-s − 4.53e6·26-s + 2.93e6·27-s − 2.50e6·29-s + 2.73e6·30-s + 7.91e6·31-s + 8.27e6·32-s + ⋯
L(s)  = 1  − 1.50·2-s − 0.914·3-s + 1.27·4-s + 0.447·5-s + 1.37·6-s − 0.414·8-s − 0.163·9-s − 0.674·10-s − 0.549·11-s − 1.16·12-s + 1.29·13-s − 0.409·15-s − 0.649·16-s − 1.79·17-s + 0.246·18-s + 1.46·19-s + 0.570·20-s + 0.829·22-s − 1.10·23-s + 0.379·24-s + 0.200·25-s − 1.94·26-s + 1.06·27-s − 0.656·29-s + 0.617·30-s + 1.53·31-s + 1.39·32-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 0.50808143370.5080814337
L(12)L(\frac12) ≈\approx 0.50808143370.5080814337
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+34.1T+512T2 1 + 34.1T + 512T^{2}
3 1+128.T+1.96e4T2 1 + 128.T + 1.96e4T^{2}
11 1+2.66e4T+2.35e9T2 1 + 2.66e4T + 2.35e9T^{2}
13 1−1.32e5T+1.06e10T2 1 - 1.32e5T + 1.06e10T^{2}
17 1+6.18e5T+1.18e11T2 1 + 6.18e5T + 1.18e11T^{2}
19 1−8.29e5T+3.22e11T2 1 - 8.29e5T + 3.22e11T^{2}
23 1+1.47e6T+1.80e12T2 1 + 1.47e6T + 1.80e12T^{2}
29 1+2.50e6T+1.45e13T2 1 + 2.50e6T + 1.45e13T^{2}
31 1−7.91e6T+2.64e13T2 1 - 7.91e6T + 2.64e13T^{2}
37 1−1.58e7T+1.29e14T2 1 - 1.58e7T + 1.29e14T^{2}
41 1−2.61e5T+3.27e14T2 1 - 2.61e5T + 3.27e14T^{2}
43 1−5.00e6T+5.02e14T2 1 - 5.00e6T + 5.02e14T^{2}
47 1+3.34e5T+1.11e15T2 1 + 3.34e5T + 1.11e15T^{2}
53 1−8.05e6T+3.29e15T2 1 - 8.05e6T + 3.29e15T^{2}
59 1+1.22e8T+8.66e15T2 1 + 1.22e8T + 8.66e15T^{2}
61 1+1.02e8T+1.16e16T2 1 + 1.02e8T + 1.16e16T^{2}
67 1+1.08e7T+2.72e16T2 1 + 1.08e7T + 2.72e16T^{2}
71 1+7.47e7T+4.58e16T2 1 + 7.47e7T + 4.58e16T^{2}
73 1−1.10e8T+5.88e16T2 1 - 1.10e8T + 5.88e16T^{2}
79 1+5.30e8T+1.19e17T2 1 + 5.30e8T + 1.19e17T^{2}
83 1+1.85e8T+1.86e17T2 1 + 1.85e8T + 1.86e17T^{2}
89 1−2.72e7T+3.50e17T2 1 - 2.72e7T + 3.50e17T^{2}
97 1−4.03e8T+7.60e17T2 1 - 4.03e8T + 7.60e17T^{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

−10.47819901182460825531291975075, −9.514562608010355661716224573751, −8.663133059683176481667889887434, −7.77213109210290031553506264357, −6.53867701780348637827830992024, −5.84887344891189691512550031026, −4.51298218211041816606222639399, −2.66805699882523536228795738023, −1.42777122827974379080456778108, −0.46265917565060696656456377019, 0.46265917565060696656456377019, 1.42777122827974379080456778108, 2.66805699882523536228795738023, 4.51298218211041816606222639399, 5.84887344891189691512550031026, 6.53867701780348637827830992024, 7.77213109210290031553506264357, 8.663133059683176481667889887434, 9.514562608010355661716224573751, 10.47819901182460825531291975075

Graph of the ZZ-function along the critical line