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Properties

Label 2-13e2-1.1-c9-0-16
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  − 2.58·2-s + 184.·3-s − 505.·4-s − 581.·5-s − 475.·6-s − 1.09e4·7-s + 2.62e3·8-s + 1.42e4·9-s + 1.50e3·10-s − 7.82e4·11-s − 9.30e4·12-s + 2.81e4·14-s − 1.07e5·15-s + 2.51e5·16-s − 1.69e5·17-s − 3.67e4·18-s − 3.98e5·19-s + 2.93e5·20-s − 2.01e6·21-s + 2.02e5·22-s + 1.48e6·23-s + 4.83e5·24-s − 1.61e6·25-s − 1.00e6·27-s + 5.51e6·28-s − 2.10e6·29-s + 2.76e5·30-s + ⋯
L(s)  = 1  − 0.114·2-s + 1.31·3-s − 0.986·4-s − 0.416·5-s − 0.149·6-s − 1.71·7-s + 0.226·8-s + 0.723·9-s + 0.0474·10-s − 1.61·11-s − 1.29·12-s + 0.196·14-s − 0.546·15-s + 0.961·16-s − 0.493·17-s − 0.0825·18-s − 0.700·19-s + 0.410·20-s − 2.25·21-s + 0.183·22-s + 1.10·23-s + 0.297·24-s − 0.826·25-s − 0.362·27-s + 1.69·28-s − 0.553·29-s + 0.0623·30-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 0.74221057060.7422105706
L(12)L(\frac12) ≈\approx 0.74221057060.7422105706
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+2.58T+512T2 1 + 2.58T + 512T^{2}
3 1−184.T+1.96e4T2 1 - 184.T + 1.96e4T^{2}
5 1+581.T+1.95e6T2 1 + 581.T + 1.95e6T^{2}
7 1+1.09e4T+4.03e7T2 1 + 1.09e4T + 4.03e7T^{2}
11 1+7.82e4T+2.35e9T2 1 + 7.82e4T + 2.35e9T^{2}
17 1+1.69e5T+1.18e11T2 1 + 1.69e5T + 1.18e11T^{2}
19 1+3.98e5T+3.22e11T2 1 + 3.98e5T + 3.22e11T^{2}
23 1−1.48e6T+1.80e12T2 1 - 1.48e6T + 1.80e12T^{2}
29 1+2.10e6T+1.45e13T2 1 + 2.10e6T + 1.45e13T^{2}
31 1−1.23e6T+2.64e13T2 1 - 1.23e6T + 2.64e13T^{2}
37 1−1.48e5T+1.29e14T2 1 - 1.48e5T + 1.29e14T^{2}
41 1−3.23e7T+3.27e14T2 1 - 3.23e7T + 3.27e14T^{2}
43 1−2.06e7T+5.02e14T2 1 - 2.06e7T + 5.02e14T^{2}
47 1+3.31e7T+1.11e15T2 1 + 3.31e7T + 1.11e15T^{2}
53 1−4.81e7T+3.29e15T2 1 - 4.81e7T + 3.29e15T^{2}
59 1+7.37e6T+8.66e15T2 1 + 7.37e6T + 8.66e15T^{2}
61 1+3.78e7T+1.16e16T2 1 + 3.78e7T + 1.16e16T^{2}
67 1+8.03e7T+2.72e16T2 1 + 8.03e7T + 2.72e16T^{2}
71 1+4.97e7T+4.58e16T2 1 + 4.97e7T + 4.58e16T^{2}
73 1−7.08e6T+5.88e16T2 1 - 7.08e6T + 5.88e16T^{2}
79 1−1.85e8T+1.19e17T2 1 - 1.85e8T + 1.19e17T^{2}
83 1+2.38e8T+1.86e17T2 1 + 2.38e8T + 1.86e17T^{2}
89 1+3.02e6T+3.50e17T2 1 + 3.02e6T + 3.50e17T^{2}
97 1−5.53e8T+7.60e17T2 1 - 5.53e8T + 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

−10.74345090369188386253704944540, −9.727502121244043185298023839771, −9.103395319637289925129724990190, −8.192118360098464470372977702853, −7.32076923405487481390635352752, −5.77395634935902324919506736930, −4.28055985509102495840818449525, −3.29511391414836595803232550514, −2.49593792877133812964702100685, −0.38516062550432490548226981807, 0.38516062550432490548226981807, 2.49593792877133812964702100685, 3.29511391414836595803232550514, 4.28055985509102495840818449525, 5.77395634935902324919506736930, 7.32076923405487481390635352752, 8.192118360098464470372977702853, 9.103395319637289925129724990190, 9.727502121244043185298023839771, 10.74345090369188386253704944540

Graph of the ZZ-function along the critical line