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Properties

Label 2-13e2-1.1-c9-0-27
Degree 22
Conductor 169169
Sign −1-1
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 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 40.6·2-s − 254.·3-s + 1.14e3·4-s − 1.76e3·5-s + 1.03e4·6-s + 1.27e3·7-s − 2.56e4·8-s + 4.48e4·9-s + 7.16e4·10-s − 5.26e4·11-s − 2.90e5·12-s − 5.20e4·14-s + 4.47e5·15-s + 4.56e5·16-s − 2.23e5·17-s − 1.82e6·18-s − 5.45e5·19-s − 2.01e6·20-s − 3.25e5·21-s + 2.13e6·22-s − 1.10e6·23-s + 6.50e6·24-s + 1.15e6·25-s − 6.40e6·27-s + 1.46e6·28-s − 1.27e6·29-s − 1.82e7·30-s + ⋯
L(s)  = 1  − 1.79·2-s − 1.81·3-s + 2.22·4-s − 1.26·5-s + 3.25·6-s + 0.201·7-s − 2.20·8-s + 2.28·9-s + 2.26·10-s − 1.08·11-s − 4.03·12-s − 0.361·14-s + 2.28·15-s + 1.74·16-s − 0.648·17-s − 4.09·18-s − 0.959·19-s − 2.81·20-s − 0.364·21-s + 1.94·22-s − 0.823·23-s + 4.00·24-s + 0.589·25-s − 2.31·27-s + 0.449·28-s − 0.335·29-s − 4.10·30-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: −1-1
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: 11
Selberg data: (2, 169, ( :9/2), −1)(2,\ 169,\ (\ :9/2),\ -1)

Particular Values

L(5)L(5) == 00
L(12)L(\frac12) == 00
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+40.6T+512T2 1 + 40.6T + 512T^{2}
3 1+254.T+1.96e4T2 1 + 254.T + 1.96e4T^{2}
5 1+1.76e3T+1.95e6T2 1 + 1.76e3T + 1.95e6T^{2}
7 1−1.27e3T+4.03e7T2 1 - 1.27e3T + 4.03e7T^{2}
11 1+5.26e4T+2.35e9T2 1 + 5.26e4T + 2.35e9T^{2}
17 1+2.23e5T+1.18e11T2 1 + 2.23e5T + 1.18e11T^{2}
19 1+5.45e5T+3.22e11T2 1 + 5.45e5T + 3.22e11T^{2}
23 1+1.10e6T+1.80e12T2 1 + 1.10e6T + 1.80e12T^{2}
29 1+1.27e6T+1.45e13T2 1 + 1.27e6T + 1.45e13T^{2}
31 1+7.41e6T+2.64e13T2 1 + 7.41e6T + 2.64e13T^{2}
37 1−4.37e6T+1.29e14T2 1 - 4.37e6T + 1.29e14T^{2}
41 1+1.41e7T+3.27e14T2 1 + 1.41e7T + 3.27e14T^{2}
43 1−2.97e7T+5.02e14T2 1 - 2.97e7T + 5.02e14T^{2}
47 1−1.62e7T+1.11e15T2 1 - 1.62e7T + 1.11e15T^{2}
53 1−8.73e7T+3.29e15T2 1 - 8.73e7T + 3.29e15T^{2}
59 1+1.01e8T+8.66e15T2 1 + 1.01e8T + 8.66e15T^{2}
61 1−2.03e8T+1.16e16T2 1 - 2.03e8T + 1.16e16T^{2}
67 1−1.43e8T+2.72e16T2 1 - 1.43e8T + 2.72e16T^{2}
71 1+6.26e7T+4.58e16T2 1 + 6.26e7T + 4.58e16T^{2}
73 1−3.27e8T+5.88e16T2 1 - 3.27e8T + 5.88e16T^{2}
79 1−2.96e8T+1.19e17T2 1 - 2.96e8T + 1.19e17T^{2}
83 1+6.46e8T+1.86e17T2 1 + 6.46e8T + 1.86e17T^{2}
89 1+5.31e8T+3.50e17T2 1 + 5.31e8T + 3.50e17T^{2}
97 1+8.88e8T+7.60e17T2 1 + 8.88e8T + 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

−10.89414842257844439961049681599, −9.830447285552277227132233124694, −8.412009732832923828051355973849, −7.54631147620291118977261153187, −6.77591111575741351450941866386, −5.57897680379157977825218894289, −4.17949615239517128010638071684, −2.02824436131446940760711689145, −0.59067012967632484528446641233, 0, 0.59067012967632484528446641233, 2.02824436131446940760711689145, 4.17949615239517128010638071684, 5.57897680379157977825218894289, 6.77591111575741351450941866386, 7.54631147620291118977261153187, 8.412009732832923828051355973849, 9.830447285552277227132233124694, 10.89414842257844439961049681599

Graph of the ZZ-function along the critical line