Properties

Label 2-17e2-1.1-c9-0-182
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

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 18.5·2-s + 248.·3-s − 166.·4-s + 410.·5-s + 4.61e3·6-s − 5.15e3·7-s − 1.26e4·8-s + 4.19e4·9-s + 7.62e3·10-s − 3.60e4·11-s − 4.12e4·12-s + 1.26e5·13-s − 9.58e4·14-s + 1.01e5·15-s − 1.49e5·16-s + 7.79e5·18-s − 4.12e5·19-s − 6.82e4·20-s − 1.27e6·21-s − 6.70e5·22-s − 2.06e6·23-s − 3.13e6·24-s − 1.78e6·25-s + 2.34e6·26-s + 5.51e6·27-s + 8.57e5·28-s + 6.64e6·29-s + ⋯
L(s)  = 1  + 0.821·2-s + 1.76·3-s − 0.324·4-s + 0.293·5-s + 1.45·6-s − 0.811·7-s − 1.08·8-s + 2.12·9-s + 0.241·10-s − 0.742·11-s − 0.574·12-s + 1.22·13-s − 0.666·14-s + 0.519·15-s − 0.569·16-s + 1.74·18-s − 0.726·19-s − 0.0953·20-s − 1.43·21-s − 0.610·22-s − 1.53·23-s − 1.92·24-s − 0.913·25-s + 1.00·26-s + 1.99·27-s + 0.263·28-s + 1.74·29-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 \( 1 - 18.5T + 512T^{2} \)
3 \( 1 - 248.T + 1.96e4T^{2} \)
5 \( 1 - 410.T + 1.95e6T^{2} \)
7 \( 1 + 5.15e3T + 4.03e7T^{2} \)
11 \( 1 + 3.60e4T + 2.35e9T^{2} \)
13 \( 1 - 1.26e5T + 1.06e10T^{2} \)
19 \( 1 + 4.12e5T + 3.22e11T^{2} \)
23 \( 1 + 2.06e6T + 1.80e12T^{2} \)
29 \( 1 - 6.64e6T + 1.45e13T^{2} \)
31 \( 1 + 3.55e6T + 2.64e13T^{2} \)
37 \( 1 + 1.09e7T + 1.29e14T^{2} \)
41 \( 1 - 2.24e6T + 3.27e14T^{2} \)
43 \( 1 + 3.86e7T + 5.02e14T^{2} \)
47 \( 1 + 3.45e7T + 1.11e15T^{2} \)
53 \( 1 - 9.58e7T + 3.29e15T^{2} \)
59 \( 1 - 7.76e7T + 8.66e15T^{2} \)
61 \( 1 + 1.94e7T + 1.16e16T^{2} \)
67 \( 1 + 2.59e8T + 2.72e16T^{2} \)
71 \( 1 - 1.12e8T + 4.58e16T^{2} \)
73 \( 1 + 1.57e8T + 5.88e16T^{2} \)
79 \( 1 + 5.78e8T + 1.19e17T^{2} \)
83 \( 1 - 1.91e8T + 1.86e17T^{2} \)
89 \( 1 + 8.16e8T + 3.50e17T^{2} \)
97 \( 1 + 1.16e9T + 7.60e17T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.712169245555303637444758209587, −8.574996581354149744839970737571, −8.270313967202470689791615308228, −6.75801958792883155153991380209, −5.76050029600979339442253617297, −4.32050917839735579841390555402, −3.56970427283641424558112413891, −2.83908134835923674740030567101, −1.76207790587087321662559535873, 0, 1.76207790587087321662559535873, 2.83908134835923674740030567101, 3.56970427283641424558112413891, 4.32050917839735579841390555402, 5.76050029600979339442253617297, 6.75801958792883155153991380209, 8.270313967202470689791615308228, 8.574996581354149744839970737571, 9.712169245555303637444758209587

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