Properties

Label 2-17e2-1.1-c9-0-87
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  − 16.2·2-s − 276.·3-s − 248.·4-s + 1.63e3·5-s + 4.48e3·6-s − 8.49e3·7-s + 1.23e4·8-s + 5.66e4·9-s − 2.65e4·10-s − 4.61e4·11-s + 6.86e4·12-s + 4.70e4·13-s + 1.37e5·14-s − 4.52e5·15-s − 7.31e4·16-s − 9.18e5·18-s + 1.92e5·19-s − 4.07e5·20-s + 2.34e6·21-s + 7.49e5·22-s − 1.30e6·23-s − 3.40e6·24-s + 7.31e5·25-s − 7.64e5·26-s − 1.01e7·27-s + 2.11e6·28-s − 3.14e6·29-s + ⋯
L(s)  = 1  − 0.717·2-s − 1.96·3-s − 0.485·4-s + 1.17·5-s + 1.41·6-s − 1.33·7-s + 1.06·8-s + 2.87·9-s − 0.840·10-s − 0.951·11-s + 0.955·12-s + 0.457·13-s + 0.959·14-s − 2.30·15-s − 0.278·16-s − 2.06·18-s + 0.338·19-s − 0.569·20-s + 2.63·21-s + 0.682·22-s − 0.975·23-s − 2.09·24-s + 0.374·25-s − 0.328·26-s − 3.69·27-s + 0.649·28-s − 0.826·29-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 \( 1 + 16.2T + 512T^{2} \)
3 \( 1 + 276.T + 1.96e4T^{2} \)
5 \( 1 - 1.63e3T + 1.95e6T^{2} \)
7 \( 1 + 8.49e3T + 4.03e7T^{2} \)
11 \( 1 + 4.61e4T + 2.35e9T^{2} \)
13 \( 1 - 4.70e4T + 1.06e10T^{2} \)
19 \( 1 - 1.92e5T + 3.22e11T^{2} \)
23 \( 1 + 1.30e6T + 1.80e12T^{2} \)
29 \( 1 + 3.14e6T + 1.45e13T^{2} \)
31 \( 1 + 3.13e6T + 2.64e13T^{2} \)
37 \( 1 - 7.07e6T + 1.29e14T^{2} \)
41 \( 1 + 3.58e6T + 3.27e14T^{2} \)
43 \( 1 - 2.56e7T + 5.02e14T^{2} \)
47 \( 1 + 4.61e7T + 1.11e15T^{2} \)
53 \( 1 - 7.23e7T + 3.29e15T^{2} \)
59 \( 1 + 6.44e7T + 8.66e15T^{2} \)
61 \( 1 - 1.13e8T + 1.16e16T^{2} \)
67 \( 1 - 3.92e7T + 2.72e16T^{2} \)
71 \( 1 - 8.04e7T + 4.58e16T^{2} \)
73 \( 1 - 2.91e8T + 5.88e16T^{2} \)
79 \( 1 - 5.23e8T + 1.19e17T^{2} \)
83 \( 1 + 3.13e8T + 1.86e17T^{2} \)
89 \( 1 + 6.56e8T + 3.50e17T^{2} \)
97 \( 1 + 1.11e8T + 7.60e17T^{2} \)
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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

−9.921207035114298280847949464975, −9.342962324103960987618830595487, −7.67168793510974558481510795976, −6.59138238087115612162870208696, −5.80435379579930130667363628346, −5.20558731451409073499054045095, −3.90496371018706345552682385027, −1.92660410463037962832322610525, −0.76379891118299267717998595406, 0, 0.76379891118299267717998595406, 1.92660410463037962832322610525, 3.90496371018706345552682385027, 5.20558731451409073499054045095, 5.80435379579930130667363628346, 6.59138238087115612162870208696, 7.67168793510974558481510795976, 9.342962324103960987618830595487, 9.921207035114298280847949464975

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