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Properties

Label 2-13e2-1.1-c9-0-29
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  − 8.72·2-s + 168.·3-s − 435.·4-s − 932.·5-s − 1.46e3·6-s − 1.29e3·7-s + 8.27e3·8-s + 8.62e3·9-s + 8.13e3·10-s + 2.52e4·11-s − 7.33e4·12-s + 1.12e4·14-s − 1.56e5·15-s + 1.50e5·16-s + 1.24e5·17-s − 7.53e4·18-s + 8.52e4·19-s + 4.06e5·20-s − 2.17e5·21-s − 2.20e5·22-s − 2.01e6·23-s + 1.39e6·24-s − 1.08e6·25-s − 1.85e6·27-s + 5.63e5·28-s − 8.38e5·29-s + 1.36e6·30-s + ⋯
L(s)  = 1  − 0.385·2-s + 1.19·3-s − 0.851·4-s − 0.667·5-s − 0.462·6-s − 0.203·7-s + 0.714·8-s + 0.438·9-s + 0.257·10-s + 0.520·11-s − 1.02·12-s + 0.0784·14-s − 0.800·15-s + 0.575·16-s + 0.361·17-s − 0.169·18-s + 0.150·19-s + 0.568·20-s − 0.244·21-s − 0.200·22-s − 1.50·23-s + 0.856·24-s − 0.554·25-s − 0.673·27-s + 0.173·28-s − 0.220·29-s + 0.308·30-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 1.5365994601.536599460
L(12)L(\frac12) ≈\approx 1.5365994601.536599460
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+8.72T+512T2 1 + 8.72T + 512T^{2}
3 1−168.T+1.96e4T2 1 - 168.T + 1.96e4T^{2}
5 1+932.T+1.95e6T2 1 + 932.T + 1.95e6T^{2}
7 1+1.29e3T+4.03e7T2 1 + 1.29e3T + 4.03e7T^{2}
11 1−2.52e4T+2.35e9T2 1 - 2.52e4T + 2.35e9T^{2}
17 1−1.24e5T+1.18e11T2 1 - 1.24e5T + 1.18e11T^{2}
19 1−8.52e4T+3.22e11T2 1 - 8.52e4T + 3.22e11T^{2}
23 1+2.01e6T+1.80e12T2 1 + 2.01e6T + 1.80e12T^{2}
29 1+8.38e5T+1.45e13T2 1 + 8.38e5T + 1.45e13T^{2}
31 1+9.11e6T+2.64e13T2 1 + 9.11e6T + 2.64e13T^{2}
37 1−1.53e7T+1.29e14T2 1 - 1.53e7T + 1.29e14T^{2}
41 1−1.91e7T+3.27e14T2 1 - 1.91e7T + 3.27e14T^{2}
43 1−3.30e7T+5.02e14T2 1 - 3.30e7T + 5.02e14T^{2}
47 1−6.34e7T+1.11e15T2 1 - 6.34e7T + 1.11e15T^{2}
53 1+4.15e7T+3.29e15T2 1 + 4.15e7T + 3.29e15T^{2}
59 1+1.67e7T+8.66e15T2 1 + 1.67e7T + 8.66e15T^{2}
61 1−8.56e7T+1.16e16T2 1 - 8.56e7T + 1.16e16T^{2}
67 1−4.09e7T+2.72e16T2 1 - 4.09e7T + 2.72e16T^{2}
71 1−1.55e8T+4.58e16T2 1 - 1.55e8T + 4.58e16T^{2}
73 1−8.73e7T+5.88e16T2 1 - 8.73e7T + 5.88e16T^{2}
79 1−6.00e8T+1.19e17T2 1 - 6.00e8T + 1.19e17T^{2}
83 1−3.65e8T+1.86e17T2 1 - 3.65e8T + 1.86e17T^{2}
89 1−3.32e8T+3.50e17T2 1 - 3.32e8T + 3.50e17T^{2}
97 1−5.38e8T+7.60e17T2 1 - 5.38e8T + 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

−10.93564394011021011287945399484, −9.572850659503106285067898440247, −9.170444432132244799119253973655, −8.007832525373815844564178552598, −7.60002368431330621634420818848, −5.78951074282403506430015212874, −4.14651791784056538683534799967, −3.59779288521559376191542521435, −2.09884317050220201249957420612, −0.61507726220777679110302580368, 0.61507726220777679110302580368, 2.09884317050220201249957420612, 3.59779288521559376191542521435, 4.14651791784056538683534799967, 5.78951074282403506430015212874, 7.60002368431330621634420818848, 8.007832525373815844564178552598, 9.170444432132244799119253973655, 9.572850659503106285067898440247, 10.93564394011021011287945399484

Graph of the ZZ-function along the critical line