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Properties

Label 2-13e2-1.1-c9-0-57
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  − 28.8·2-s − 204.·3-s + 317.·4-s + 258.·5-s + 5.89e3·6-s + 8.86e3·7-s + 5.59e3·8-s + 2.21e4·9-s − 7.45e3·10-s − 3.60e4·11-s − 6.49e4·12-s − 2.55e5·14-s − 5.29e4·15-s − 3.23e5·16-s + 3.27e5·17-s − 6.38e5·18-s − 2.65e5·19-s + 8.22e4·20-s − 1.81e6·21-s + 1.03e6·22-s − 2.42e6·23-s − 1.14e6·24-s − 1.88e6·25-s − 5.09e5·27-s + 2.81e6·28-s + 3.99e6·29-s + 1.52e6·30-s + ⋯
L(s)  = 1  − 1.27·2-s − 1.45·3-s + 0.620·4-s + 0.185·5-s + 1.85·6-s + 1.39·7-s + 0.483·8-s + 1.12·9-s − 0.235·10-s − 0.743·11-s − 0.904·12-s − 1.77·14-s − 0.270·15-s − 1.23·16-s + 0.950·17-s − 1.43·18-s − 0.467·19-s + 0.114·20-s − 2.03·21-s + 0.946·22-s − 1.80·23-s − 0.704·24-s − 0.965·25-s − 0.184·27-s + 0.865·28-s + 1.04·29-s + 0.343·30-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+28.8T+512T2 1 + 28.8T + 512T^{2}
3 1+204.T+1.96e4T2 1 + 204.T + 1.96e4T^{2}
5 1−258.T+1.95e6T2 1 - 258.T + 1.95e6T^{2}
7 1−8.86e3T+4.03e7T2 1 - 8.86e3T + 4.03e7T^{2}
11 1+3.60e4T+2.35e9T2 1 + 3.60e4T + 2.35e9T^{2}
17 1−3.27e5T+1.18e11T2 1 - 3.27e5T + 1.18e11T^{2}
19 1+2.65e5T+3.22e11T2 1 + 2.65e5T + 3.22e11T^{2}
23 1+2.42e6T+1.80e12T2 1 + 2.42e6T + 1.80e12T^{2}
29 1−3.99e6T+1.45e13T2 1 - 3.99e6T + 1.45e13T^{2}
31 1−6.45e6T+2.64e13T2 1 - 6.45e6T + 2.64e13T^{2}
37 1−8.15e6T+1.29e14T2 1 - 8.15e6T + 1.29e14T^{2}
41 1−7.20e5T+3.27e14T2 1 - 7.20e5T + 3.27e14T^{2}
43 1+4.13e7T+5.02e14T2 1 + 4.13e7T + 5.02e14T^{2}
47 1+3.67e7T+1.11e15T2 1 + 3.67e7T + 1.11e15T^{2}
53 1−1.67e7T+3.29e15T2 1 - 1.67e7T + 3.29e15T^{2}
59 1−5.89e7T+8.66e15T2 1 - 5.89e7T + 8.66e15T^{2}
61 1−1.28e8T+1.16e16T2 1 - 1.28e8T + 1.16e16T^{2}
67 1−1.91e8T+2.72e16T2 1 - 1.91e8T + 2.72e16T^{2}
71 1+3.40e8T+4.58e16T2 1 + 3.40e8T + 4.58e16T^{2}
73 1+3.19e8T+5.88e16T2 1 + 3.19e8T + 5.88e16T^{2}
79 1−2.44e8T+1.19e17T2 1 - 2.44e8T + 1.19e17T^{2}
83 1−4.38e8T+1.86e17T2 1 - 4.38e8T + 1.86e17T^{2}
89 1−7.90e8T+3.50e17T2 1 - 7.90e8T + 3.50e17T^{2}
97 1−1.65e8T+7.60e17T2 1 - 1.65e8T + 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

−10.30815623236519368221762925635, −10.06033972625015681480168642870, −8.286569791887793887122890143708, −7.86354074932348836783735663476, −6.43984036369092908324497654228, −5.31972382649754524275630073508, −4.46347658996413836047935326627, −2.00955452680269101512066702834, −1.00789806563369914359942812168, 0, 1.00789806563369914359942812168, 2.00955452680269101512066702834, 4.46347658996413836047935326627, 5.31972382649754524275630073508, 6.43984036369092908324497654228, 7.86354074932348836783735663476, 8.286569791887793887122890143708, 10.06033972625015681480168642870, 10.30815623236519368221762925635

Graph of the ZZ-function along the critical line