Properties

Label 2-17e2-1.1-c9-0-173
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  + 39.9·2-s − 192.·3-s + 1.08e3·4-s + 2.27e3·5-s − 7.67e3·6-s − 1.05e4·7-s + 2.27e4·8-s + 1.72e4·9-s + 9.06e4·10-s − 2.61e4·11-s − 2.07e5·12-s + 1.30e5·13-s − 4.20e5·14-s − 4.36e5·15-s + 3.55e5·16-s + 6.88e5·18-s − 4.46e5·19-s + 2.45e6·20-s + 2.02e6·21-s − 1.04e6·22-s + 3.40e5·23-s − 4.37e6·24-s + 3.19e6·25-s + 5.21e6·26-s + 4.69e5·27-s − 1.13e7·28-s − 3.01e6·29-s + ⋯
L(s)  = 1  + 1.76·2-s − 1.36·3-s + 2.11·4-s + 1.62·5-s − 2.41·6-s − 1.65·7-s + 1.96·8-s + 0.875·9-s + 2.86·10-s − 0.539·11-s − 2.89·12-s + 1.26·13-s − 2.92·14-s − 2.22·15-s + 1.35·16-s + 1.54·18-s − 0.785·19-s + 3.43·20-s + 2.26·21-s − 0.951·22-s + 0.253·23-s − 2.69·24-s + 1.63·25-s + 2.23·26-s + 0.169·27-s − 3.50·28-s − 0.791·29-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 \( 1 - 39.9T + 512T^{2} \)
3 \( 1 + 192.T + 1.96e4T^{2} \)
5 \( 1 - 2.27e3T + 1.95e6T^{2} \)
7 \( 1 + 1.05e4T + 4.03e7T^{2} \)
11 \( 1 + 2.61e4T + 2.35e9T^{2} \)
13 \( 1 - 1.30e5T + 1.06e10T^{2} \)
19 \( 1 + 4.46e5T + 3.22e11T^{2} \)
23 \( 1 - 3.40e5T + 1.80e12T^{2} \)
29 \( 1 + 3.01e6T + 1.45e13T^{2} \)
31 \( 1 + 1.67e6T + 2.64e13T^{2} \)
37 \( 1 + 1.57e7T + 1.29e14T^{2} \)
41 \( 1 + 1.68e7T + 3.27e14T^{2} \)
43 \( 1 + 4.62e6T + 5.02e14T^{2} \)
47 \( 1 + 5.82e6T + 1.11e15T^{2} \)
53 \( 1 - 2.06e7T + 3.29e15T^{2} \)
59 \( 1 + 5.56e7T + 8.66e15T^{2} \)
61 \( 1 - 1.23e8T + 1.16e16T^{2} \)
67 \( 1 + 1.03e8T + 2.72e16T^{2} \)
71 \( 1 + 2.91e8T + 4.58e16T^{2} \)
73 \( 1 + 2.25e8T + 5.88e16T^{2} \)
79 \( 1 - 6.19e8T + 1.19e17T^{2} \)
83 \( 1 - 1.60e8T + 1.86e17T^{2} \)
89 \( 1 - 6.09e8T + 3.50e17T^{2} \)
97 \( 1 + 5.75e8T + 7.60e17T^{2} \)
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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

−10.35301708317438315258851917722, −9.097717379678667611409743275192, −6.75656441401373735798732391535, −6.40956523811360161698860746919, −5.70581526867285653560107625818, −5.20471460962266488434676624680, −3.78044186893670117926778713600, −2.76147693157195181896745740988, −1.57823795559131246156841480371, 0, 1.57823795559131246156841480371, 2.76147693157195181896745740988, 3.78044186893670117926778713600, 5.20471460962266488434676624680, 5.70581526867285653560107625818, 6.40956523811360161698860746919, 6.75656441401373735798732391535, 9.097717379678667611409743275192, 10.35301708317438315258851917722

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