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Properties

Label 2-13e2-1.1-c9-0-87
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  + 43.0·2-s − 29.4·3-s + 1.33e3·4-s + 1.66e3·5-s − 1.26e3·6-s + 1.04e4·7-s + 3.56e4·8-s − 1.88e4·9-s + 7.16e4·10-s + 3.86e4·11-s − 3.95e4·12-s + 4.51e5·14-s − 4.91e4·15-s + 8.46e5·16-s − 1.60e4·17-s − 8.09e5·18-s − 1.88e5·19-s + 2.23e6·20-s − 3.09e5·21-s + 1.66e6·22-s + 9.28e5·23-s − 1.05e6·24-s + 8.22e5·25-s + 1.13e6·27-s + 1.40e7·28-s − 7.11e6·29-s − 2.11e6·30-s + ⋯
L(s)  = 1  + 1.90·2-s − 0.210·3-s + 2.61·4-s + 1.19·5-s − 0.399·6-s + 1.65·7-s + 3.07·8-s − 0.955·9-s + 2.26·10-s + 0.795·11-s − 0.549·12-s + 3.14·14-s − 0.250·15-s + 3.23·16-s − 0.0467·17-s − 1.81·18-s − 0.331·19-s + 3.11·20-s − 0.347·21-s + 1.51·22-s + 0.691·23-s − 0.646·24-s + 0.421·25-s + 0.411·27-s + 4.32·28-s − 1.86·29-s − 0.476·30-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 10.8822608010.88226080
L(12)L(\frac12) ≈\approx 10.8822608010.88226080
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 1−43.0T+512T2 1 - 43.0T + 512T^{2}
3 1+29.4T+1.96e4T2 1 + 29.4T + 1.96e4T^{2}
5 1−1.66e3T+1.95e6T2 1 - 1.66e3T + 1.95e6T^{2}
7 1−1.04e4T+4.03e7T2 1 - 1.04e4T + 4.03e7T^{2}
11 1−3.86e4T+2.35e9T2 1 - 3.86e4T + 2.35e9T^{2}
17 1+1.60e4T+1.18e11T2 1 + 1.60e4T + 1.18e11T^{2}
19 1+1.88e5T+3.22e11T2 1 + 1.88e5T + 3.22e11T^{2}
23 1−9.28e5T+1.80e12T2 1 - 9.28e5T + 1.80e12T^{2}
29 1+7.11e6T+1.45e13T2 1 + 7.11e6T + 1.45e13T^{2}
31 1+6.35e6T+2.64e13T2 1 + 6.35e6T + 2.64e13T^{2}
37 1+1.05e7T+1.29e14T2 1 + 1.05e7T + 1.29e14T^{2}
41 1−1.59e7T+3.27e14T2 1 - 1.59e7T + 3.27e14T^{2}
43 1+1.14e7T+5.02e14T2 1 + 1.14e7T + 5.02e14T^{2}
47 1+2.91e7T+1.11e15T2 1 + 2.91e7T + 1.11e15T^{2}
53 1−3.86e7T+3.29e15T2 1 - 3.86e7T + 3.29e15T^{2}
59 1+1.28e8T+8.66e15T2 1 + 1.28e8T + 8.66e15T^{2}
61 1+7.27e7T+1.16e16T2 1 + 7.27e7T + 1.16e16T^{2}
67 1+8.28e7T+2.72e16T2 1 + 8.28e7T + 2.72e16T^{2}
71 1−1.68e8T+4.58e16T2 1 - 1.68e8T + 4.58e16T^{2}
73 1−2.83e8T+5.88e16T2 1 - 2.83e8T + 5.88e16T^{2}
79 1−5.18e8T+1.19e17T2 1 - 5.18e8T + 1.19e17T^{2}
83 1+2.25e8T+1.86e17T2 1 + 2.25e8T + 1.86e17T^{2}
89 1−8.31e8T+3.50e17T2 1 - 8.31e8T + 3.50e17T^{2}
97 1−2.66e8T+7.60e17T2 1 - 2.66e8T + 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

−11.25359982842790023464961566087, −10.82887407773419190931681208982, −9.068030584528098448513207619676, −7.59369549117020591009894716092, −6.35183655354116614571921193816, −5.50543054222555937821239523695, −4.92562581213344665502857937526, −3.63048073764595871870774479781, −2.18840618909195532106970022554, −1.56622966613251818465566963611, 1.56622966613251818465566963611, 2.18840618909195532106970022554, 3.63048073764595871870774479781, 4.92562581213344665502857937526, 5.50543054222555937821239523695, 6.35183655354116614571921193816, 7.59369549117020591009894716092, 9.068030584528098448513207619676, 10.82887407773419190931681208982, 11.25359982842790023464961566087

Graph of the ZZ-function along the critical line