Properties

Label 2-17e2-1.1-c9-0-52
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 37.3·2-s − 245.·3-s + 884.·4-s − 1.54e3·5-s + 9.18e3·6-s − 9.37e3·7-s − 1.39e4·8-s + 4.07e4·9-s + 5.79e4·10-s − 3.18e4·11-s − 2.17e5·12-s − 3.83e3·13-s + 3.50e5·14-s + 3.80e5·15-s + 6.75e4·16-s − 1.52e6·18-s + 7.04e5·19-s − 1.37e6·20-s + 2.30e6·21-s + 1.19e6·22-s − 1.41e5·23-s + 3.42e6·24-s + 4.47e5·25-s + 1.43e5·26-s − 5.16e6·27-s − 8.29e6·28-s + 5.40e6·29-s + ⋯
L(s)  = 1  − 1.65·2-s − 1.75·3-s + 1.72·4-s − 1.10·5-s + 2.89·6-s − 1.47·7-s − 1.20·8-s + 2.06·9-s + 1.83·10-s − 0.655·11-s − 3.02·12-s − 0.0372·13-s + 2.43·14-s + 1.94·15-s + 0.257·16-s − 3.41·18-s + 1.24·19-s − 1.91·20-s + 2.58·21-s + 1.08·22-s − 0.105·23-s + 2.10·24-s + 0.228·25-s + 0.0615·26-s − 1.87·27-s − 2.55·28-s + 1.41·29-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: \(0\)
Selberg data: \((2,\ 289,\ (\ :9/2),\ 1)\)

Particular Values

\(L(5)\) \(\approx\) \(0.3019754114\)
\(L(\frac12)\) \(\approx\) \(0.3019754114\)
\(L(\frac{11}{2})\) 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 \( 1 + 37.3T + 512T^{2} \)
3 \( 1 + 245.T + 1.96e4T^{2} \)
5 \( 1 + 1.54e3T + 1.95e6T^{2} \)
7 \( 1 + 9.37e3T + 4.03e7T^{2} \)
11 \( 1 + 3.18e4T + 2.35e9T^{2} \)
13 \( 1 + 3.83e3T + 1.06e10T^{2} \)
19 \( 1 - 7.04e5T + 3.22e11T^{2} \)
23 \( 1 + 1.41e5T + 1.80e12T^{2} \)
29 \( 1 - 5.40e6T + 1.45e13T^{2} \)
31 \( 1 - 8.78e6T + 2.64e13T^{2} \)
37 \( 1 + 2.94e5T + 1.29e14T^{2} \)
41 \( 1 - 1.41e7T + 3.27e14T^{2} \)
43 \( 1 - 1.87e7T + 5.02e14T^{2} \)
47 \( 1 - 2.19e7T + 1.11e15T^{2} \)
53 \( 1 - 8.33e7T + 3.29e15T^{2} \)
59 \( 1 - 1.05e8T + 8.66e15T^{2} \)
61 \( 1 - 9.79e6T + 1.16e16T^{2} \)
67 \( 1 - 1.49e8T + 2.72e16T^{2} \)
71 \( 1 + 7.01e6T + 4.58e16T^{2} \)
73 \( 1 + 9.11e7T + 5.88e16T^{2} \)
79 \( 1 + 8.25e7T + 1.19e17T^{2} \)
83 \( 1 + 5.73e7T + 1.86e17T^{2} \)
89 \( 1 + 6.66e7T + 3.50e17T^{2} \)
97 \( 1 + 5.74e8T + 7.60e17T^{2} \)
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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

−10.18024027093566446868163499932, −9.638239752143314428964034105701, −8.269332544748013010180586812200, −7.27892934675817693227127772390, −6.68453422245624334868227114593, −5.67729045167157546984306396631, −4.29424022114903160804906009698, −2.79538526909156345803401728809, −0.797341094327779799519936849970, −0.55580749060222986739554207473, 0.55580749060222986739554207473, 0.797341094327779799519936849970, 2.79538526909156345803401728809, 4.29424022114903160804906009698, 5.67729045167157546984306396631, 6.68453422245624334868227114593, 7.27892934675817693227127772390, 8.269332544748013010180586812200, 9.638239752143314428964034105701, 10.18024027093566446868163499932

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