Properties

Label 2-17e2-1.1-c9-0-162
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  − 0.561·2-s + 191.·3-s − 511.·4-s + 819.·5-s − 107.·6-s + 1.08e3·7-s + 574.·8-s + 1.70e4·9-s − 460.·10-s − 4.82e4·11-s − 9.80e4·12-s − 1.07e5·13-s − 609.·14-s + 1.57e5·15-s + 2.61e5·16-s − 9.56e3·18-s + 8.72e5·19-s − 4.19e5·20-s + 2.07e5·21-s + 2.71e4·22-s + 1.91e6·23-s + 1.10e5·24-s − 1.28e6·25-s + 6.05e4·26-s − 5.09e5·27-s − 5.55e5·28-s + 4.76e5·29-s + ⋯
L(s)  = 1  − 0.0248·2-s + 1.36·3-s − 0.999·4-s + 0.586·5-s − 0.0338·6-s + 0.170·7-s + 0.0496·8-s + 0.864·9-s − 0.0145·10-s − 0.994·11-s − 1.36·12-s − 1.04·13-s − 0.00424·14-s + 0.800·15-s + 0.998·16-s − 0.0214·18-s + 1.53·19-s − 0.586·20-s + 0.233·21-s + 0.0246·22-s + 1.42·23-s + 0.0677·24-s − 0.655·25-s + 0.0259·26-s − 0.184·27-s − 0.170·28-s + 0.125·29-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 \( 1 + 0.561T + 512T^{2} \)
3 \( 1 - 191.T + 1.96e4T^{2} \)
5 \( 1 - 819.T + 1.95e6T^{2} \)
7 \( 1 - 1.08e3T + 4.03e7T^{2} \)
11 \( 1 + 4.82e4T + 2.35e9T^{2} \)
13 \( 1 + 1.07e5T + 1.06e10T^{2} \)
19 \( 1 - 8.72e5T + 3.22e11T^{2} \)
23 \( 1 - 1.91e6T + 1.80e12T^{2} \)
29 \( 1 - 4.76e5T + 1.45e13T^{2} \)
31 \( 1 + 1.17e4T + 2.64e13T^{2} \)
37 \( 1 + 3.41e6T + 1.29e14T^{2} \)
41 \( 1 - 6.67e6T + 3.27e14T^{2} \)
43 \( 1 + 2.97e7T + 5.02e14T^{2} \)
47 \( 1 + 4.42e7T + 1.11e15T^{2} \)
53 \( 1 - 4.59e7T + 3.29e15T^{2} \)
59 \( 1 - 2.34e6T + 8.66e15T^{2} \)
61 \( 1 - 1.31e8T + 1.16e16T^{2} \)
67 \( 1 + 1.30e8T + 2.72e16T^{2} \)
71 \( 1 + 1.91e8T + 4.58e16T^{2} \)
73 \( 1 + 3.50e8T + 5.88e16T^{2} \)
79 \( 1 - 4.60e8T + 1.19e17T^{2} \)
83 \( 1 + 8.12e8T + 1.86e17T^{2} \)
89 \( 1 + 6.40e8T + 3.50e17T^{2} \)
97 \( 1 - 2.77e8T + 7.60e17T^{2} \)
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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

−9.674551286803701455714907377459, −8.860150181945039101065579263242, −8.002598529094323454850860680338, −7.26444111171989457623091313824, −5.47027423603937223106119159677, −4.79152845426682876004738698243, −3.38452368687311466140331522189, −2.65276974515352988828508263131, −1.42876497455374686906709236303, 0, 1.42876497455374686906709236303, 2.65276974515352988828508263131, 3.38452368687311466140331522189, 4.79152845426682876004738698243, 5.47027423603937223106119159677, 7.26444111171989457623091313824, 8.002598529094323454850860680338, 8.860150181945039101065579263242, 9.674551286803701455714907377459

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