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Properties

Label 2-13e2-1.1-c9-0-1
Degree 22
Conductor 169169
Sign 11
Analytic cond. 87.041087.0410
Root an. cond. 9.329579.32957
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  − 36.6·2-s − 130.·3-s + 834.·4-s + 1.41e3·5-s + 4.77e3·6-s − 7.50e3·7-s − 1.18e4·8-s − 2.77e3·9-s − 5.20e4·10-s − 4.28e4·11-s − 1.08e5·12-s + 2.75e5·14-s − 1.84e5·15-s + 6.94e3·16-s − 5.35e5·17-s + 1.01e5·18-s − 4.18e5·19-s + 1.18e6·20-s + 9.76e5·21-s + 1.57e6·22-s + 2.94e5·23-s + 1.53e6·24-s + 5.63e4·25-s + 2.92e6·27-s − 6.26e6·28-s + 1.62e6·29-s + 6.76e6·30-s + ⋯
L(s)  = 1  − 1.62·2-s − 0.926·3-s + 1.62·4-s + 1.01·5-s + 1.50·6-s − 1.18·7-s − 1.02·8-s − 0.141·9-s − 1.64·10-s − 0.882·11-s − 1.51·12-s + 1.91·14-s − 0.940·15-s + 0.0265·16-s − 1.55·17-s + 0.228·18-s − 0.737·19-s + 1.65·20-s + 1.09·21-s + 1.43·22-s + 0.219·23-s + 0.946·24-s + 0.0288·25-s + 1.05·27-s − 1.92·28-s + 0.425·29-s + 1.52·30-s + ⋯

Functional equation

Λ(s)=(169s/2 ΓC(s) L(s)=( Λ(10−s)\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}
Λ(s)=(169s/2 ΓC(s+9/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 169169    =    13213^{2}
Sign: 11
Analytic conductor: 87.041087.0410
Root analytic conductor: 9.329579.32957
Motivic weight: 99
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 169, ( :9/2), 1)(2,\ 169,\ (\ :9/2),\ 1)

Particular Values

L(5)L(5) ≈\approx 0.0025080303460.002508030346
L(12)L(\frac12) ≈\approx 0.0025080303460.002508030346
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 1
good2 1+36.6T+512T2 1 + 36.6T + 512T^{2}
3 1+130.T+1.96e4T2 1 + 130.T + 1.96e4T^{2}
5 1−1.41e3T+1.95e6T2 1 - 1.41e3T + 1.95e6T^{2}
7 1+7.50e3T+4.03e7T2 1 + 7.50e3T + 4.03e7T^{2}
11 1+4.28e4T+2.35e9T2 1 + 4.28e4T + 2.35e9T^{2}
17 1+5.35e5T+1.18e11T2 1 + 5.35e5T + 1.18e11T^{2}
19 1+4.18e5T+3.22e11T2 1 + 4.18e5T + 3.22e11T^{2}
23 1−2.94e5T+1.80e12T2 1 - 2.94e5T + 1.80e12T^{2}
29 1−1.62e6T+1.45e13T2 1 - 1.62e6T + 1.45e13T^{2}
31 1−3.25e6T+2.64e13T2 1 - 3.25e6T + 2.64e13T^{2}
37 1+2.00e7T+1.29e14T2 1 + 2.00e7T + 1.29e14T^{2}
41 1+3.44e7T+3.27e14T2 1 + 3.44e7T + 3.27e14T^{2}
43 1−3.13e6T+5.02e14T2 1 - 3.13e6T + 5.02e14T^{2}
47 1+5.82e7T+1.11e15T2 1 + 5.82e7T + 1.11e15T^{2}
53 1+2.43e7T+3.29e15T2 1 + 2.43e7T + 3.29e15T^{2}
59 1+1.61e8T+8.66e15T2 1 + 1.61e8T + 8.66e15T^{2}
61 1−8.43e7T+1.16e16T2 1 - 8.43e7T + 1.16e16T^{2}
67 1−4.89e7T+2.72e16T2 1 - 4.89e7T + 2.72e16T^{2}
71 1+3.31e7T+4.58e16T2 1 + 3.31e7T + 4.58e16T^{2}
73 1+2.22e8T+5.88e16T2 1 + 2.22e8T + 5.88e16T^{2}
79 1−1.28e7T+1.19e17T2 1 - 1.28e7T + 1.19e17T^{2}
83 1+1.32e8T+1.86e17T2 1 + 1.32e8T + 1.86e17T^{2}
89 1−5.85e8T+3.50e17T2 1 - 5.85e8T + 3.50e17T^{2}
97 1+1.03e9T+7.60e17T2 1 + 1.03e9T + 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

−10.59248526464392822198050496492, −10.18785324871698896242820873957, −9.188845475461198385387112244740, −8.343498727734344220806414949549, −6.70636143490194041816442764806, −6.37333148766390652078154920877, −5.03310622623841576448492844398, −2.79538866948085720545680150671, −1.70848428039180612854078971095, −0.03196910568340086720941339439, 0.03196910568340086720941339439, 1.70848428039180612854078971095, 2.79538866948085720545680150671, 5.03310622623841576448492844398, 6.37333148766390652078154920877, 6.70636143490194041816442764806, 8.343498727734344220806414949549, 9.188845475461198385387112244740, 10.18785324871698896242820873957, 10.59248526464392822198050496492

Graph of the ZZ-function along the critical line