Properties

Label 2-17e2-1.1-c9-0-161
Degree $2$
Conductor $289$
Sign $-1$
Analytic cond. $148.845$
Root an. cond. $12.2002$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 43.4·2-s − 157.·3-s + 1.37e3·4-s − 2.50e3·5-s − 6.82e3·6-s + 5.91e3·7-s + 3.75e4·8-s + 4.98e3·9-s − 1.08e5·10-s − 6.77e3·11-s − 2.16e5·12-s − 1.23e4·13-s + 2.57e5·14-s + 3.93e5·15-s + 9.25e5·16-s + 2.16e5·18-s + 9.46e4·19-s − 3.44e6·20-s − 9.29e5·21-s − 2.94e5·22-s + 1.61e6·23-s − 5.89e6·24-s + 4.32e6·25-s − 5.34e5·26-s + 2.30e6·27-s + 8.13e6·28-s + 4.52e5·29-s + ⋯
L(s)  = 1  + 1.92·2-s − 1.11·3-s + 2.68·4-s − 1.79·5-s − 2.14·6-s + 0.931·7-s + 3.23·8-s + 0.253·9-s − 3.44·10-s − 0.139·11-s − 3.00·12-s − 0.119·13-s + 1.78·14-s + 2.00·15-s + 3.53·16-s + 0.486·18-s + 0.166·19-s − 4.81·20-s − 1.04·21-s − 0.267·22-s + 1.20·23-s − 3.62·24-s + 2.21·25-s − 0.229·26-s + 0.836·27-s + 2.50·28-s + 0.118·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(289\)    =    \(17^{2}\)
Sign: $-1$
Analytic conductor: \(148.845\)
Root analytic conductor: \(12.2002\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{289} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 289,\ (\ :9/2),\ -1)\)

Particular Values

\(L(5)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
good2 \( 1 - 43.4T + 512T^{2} \)
3 \( 1 + 157.T + 1.96e4T^{2} \)
5 \( 1 + 2.50e3T + 1.95e6T^{2} \)
7 \( 1 - 5.91e3T + 4.03e7T^{2} \)
11 \( 1 + 6.77e3T + 2.35e9T^{2} \)
13 \( 1 + 1.23e4T + 1.06e10T^{2} \)
19 \( 1 - 9.46e4T + 3.22e11T^{2} \)
23 \( 1 - 1.61e6T + 1.80e12T^{2} \)
29 \( 1 - 4.52e5T + 1.45e13T^{2} \)
31 \( 1 + 9.43e6T + 2.64e13T^{2} \)
37 \( 1 + 5.76e6T + 1.29e14T^{2} \)
41 \( 1 + 1.78e7T + 3.27e14T^{2} \)
43 \( 1 + 1.47e7T + 5.02e14T^{2} \)
47 \( 1 + 4.18e7T + 1.11e15T^{2} \)
53 \( 1 + 2.97e7T + 3.29e15T^{2} \)
59 \( 1 - 9.64e7T + 8.66e15T^{2} \)
61 \( 1 + 6.90e7T + 1.16e16T^{2} \)
67 \( 1 - 1.04e8T + 2.72e16T^{2} \)
71 \( 1 + 9.99e5T + 4.58e16T^{2} \)
73 \( 1 + 6.92e7T + 5.88e16T^{2} \)
79 \( 1 - 1.45e8T + 1.19e17T^{2} \)
83 \( 1 + 2.99e8T + 1.86e17T^{2} \)
89 \( 1 + 9.92e8T + 3.50e17T^{2} \)
97 \( 1 - 1.69e9T + 7.60e17T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.70172811743467076347634592817, −8.310910417329726478673535499952, −7.33962608233071599443178335395, −6.66155267554513824900280077920, −5.24023152204332933463446050387, −4.94691039558342400067198789348, −3.92970329188655022974300882892, −3.04974462941169398250336394064, −1.42559945561362844322175645807, 0, 1.42559945561362844322175645807, 3.04974462941169398250336394064, 3.92970329188655022974300882892, 4.94691039558342400067198789348, 5.24023152204332933463446050387, 6.66155267554513824900280077920, 7.33962608233071599443178335395, 8.310910417329726478673535499952, 10.70172811743467076347634592817

Graph of the $Z$-function along the critical line