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Properties

Label 2-13e2-1.1-c9-0-5
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  + 25.3·2-s − 120.·3-s + 130.·4-s − 1.00e3·5-s − 3.06e3·6-s − 3.16e3·7-s − 9.66e3·8-s − 5.07e3·9-s − 2.54e4·10-s − 2.92e4·11-s − 1.58e4·12-s − 8.02e4·14-s + 1.21e5·15-s − 3.12e5·16-s − 2.27e5·17-s − 1.28e5·18-s − 4.16e5·19-s − 1.31e5·20-s + 3.82e5·21-s − 7.40e5·22-s − 2.24e6·23-s + 1.16e6·24-s − 9.44e5·25-s + 2.99e6·27-s − 4.14e5·28-s − 1.41e5·29-s + 3.07e6·30-s + ⋯
L(s)  = 1  + 1.12·2-s − 0.861·3-s + 0.255·4-s − 0.718·5-s − 0.965·6-s − 0.498·7-s − 0.834·8-s − 0.257·9-s − 0.805·10-s − 0.601·11-s − 0.220·12-s − 0.558·14-s + 0.619·15-s − 1.19·16-s − 0.661·17-s − 0.288·18-s − 0.733·19-s − 0.183·20-s + 0.429·21-s − 0.674·22-s − 1.67·23-s + 0.718·24-s − 0.483·25-s + 1.08·27-s − 0.127·28-s − 0.0371·29-s + 0.693·30-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 0.28218267230.2821826723
L(12)L(\frac12) ≈\approx 0.28218267230.2821826723
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 1−25.3T+512T2 1 - 25.3T + 512T^{2}
3 1+120.T+1.96e4T2 1 + 120.T + 1.96e4T^{2}
5 1+1.00e3T+1.95e6T2 1 + 1.00e3T + 1.95e6T^{2}
7 1+3.16e3T+4.03e7T2 1 + 3.16e3T + 4.03e7T^{2}
11 1+2.92e4T+2.35e9T2 1 + 2.92e4T + 2.35e9T^{2}
17 1+2.27e5T+1.18e11T2 1 + 2.27e5T + 1.18e11T^{2}
19 1+4.16e5T+3.22e11T2 1 + 4.16e5T + 3.22e11T^{2}
23 1+2.24e6T+1.80e12T2 1 + 2.24e6T + 1.80e12T^{2}
29 1+1.41e5T+1.45e13T2 1 + 1.41e5T + 1.45e13T^{2}
31 1+1.84e6T+2.64e13T2 1 + 1.84e6T + 2.64e13T^{2}
37 1−1.45e5T+1.29e14T2 1 - 1.45e5T + 1.29e14T^{2}
41 1−1.15e7T+3.27e14T2 1 - 1.15e7T + 3.27e14T^{2}
43 1−3.71e7T+5.02e14T2 1 - 3.71e7T + 5.02e14T^{2}
47 1+3.05e7T+1.11e15T2 1 + 3.05e7T + 1.11e15T^{2}
53 1−8.17e7T+3.29e15T2 1 - 8.17e7T + 3.29e15T^{2}
59 1+3.46e7T+8.66e15T2 1 + 3.46e7T + 8.66e15T^{2}
61 1+1.76e8T+1.16e16T2 1 + 1.76e8T + 1.16e16T^{2}
67 1−1.88e8T+2.72e16T2 1 - 1.88e8T + 2.72e16T^{2}
71 1+4.23e8T+4.58e16T2 1 + 4.23e8T + 4.58e16T^{2}
73 1+1.11e8T+5.88e16T2 1 + 1.11e8T + 5.88e16T^{2}
79 1−2.46e8T+1.19e17T2 1 - 2.46e8T + 1.19e17T^{2}
83 1−7.57e8T+1.86e17T2 1 - 7.57e8T + 1.86e17T^{2}
89 1+1.61e8T+3.50e17T2 1 + 1.61e8T + 3.50e17T^{2}
97 1−7.15e8T+7.60e17T2 1 - 7.15e8T + 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

−11.43927572820388705006874622671, −10.45223130696607767572084346673, −9.063584629926606145280838304817, −7.85323240073822197194737587843, −6.39839816197244797116064006096, −5.75599842924168714446880460173, −4.61346501265048693970798898594, −3.75580371561804240012639153415, −2.47363373035517351595004059726, −0.21672693494326634403472919929, 0.21672693494326634403472919929, 2.47363373035517351595004059726, 3.75580371561804240012639153415, 4.61346501265048693970798898594, 5.75599842924168714446880460173, 6.39839816197244797116064006096, 7.85323240073822197194737587843, 9.063584629926606145280838304817, 10.45223130696607767572084346673, 11.43927572820388705006874622671

Graph of the ZZ-function along the critical line