Properties

Label 2-17e2-1.1-c9-0-130
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  − 15.8·2-s − 48.4·3-s − 260.·4-s + 2.42e3·5-s + 769.·6-s − 4.46e3·7-s + 1.22e4·8-s − 1.73e4·9-s − 3.85e4·10-s + 2.19e4·11-s + 1.26e4·12-s + 1.43e5·13-s + 7.08e4·14-s − 1.17e5·15-s − 6.13e4·16-s + 2.75e5·18-s − 8.90e5·19-s − 6.31e5·20-s + 2.16e5·21-s − 3.48e5·22-s + 6.62e5·23-s − 5.94e5·24-s + 3.94e6·25-s − 2.27e6·26-s + 1.79e6·27-s + 1.16e6·28-s − 7.32e6·29-s + ⋯
L(s)  = 1  − 0.701·2-s − 0.345·3-s − 0.507·4-s + 1.73·5-s + 0.242·6-s − 0.703·7-s + 1.05·8-s − 0.880·9-s − 1.21·10-s + 0.452·11-s + 0.175·12-s + 1.39·13-s + 0.493·14-s − 0.600·15-s − 0.234·16-s + 0.617·18-s − 1.56·19-s − 0.882·20-s + 0.242·21-s − 0.317·22-s + 0.493·23-s − 0.365·24-s + 2.01·25-s − 0.975·26-s + 0.649·27-s + 0.357·28-s − 1.92·29-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 \( 1 + 15.8T + 512T^{2} \)
3 \( 1 + 48.4T + 1.96e4T^{2} \)
5 \( 1 - 2.42e3T + 1.95e6T^{2} \)
7 \( 1 + 4.46e3T + 4.03e7T^{2} \)
11 \( 1 - 2.19e4T + 2.35e9T^{2} \)
13 \( 1 - 1.43e5T + 1.06e10T^{2} \)
19 \( 1 + 8.90e5T + 3.22e11T^{2} \)
23 \( 1 - 6.62e5T + 1.80e12T^{2} \)
29 \( 1 + 7.32e6T + 1.45e13T^{2} \)
31 \( 1 - 2.75e6T + 2.64e13T^{2} \)
37 \( 1 - 7.46e6T + 1.29e14T^{2} \)
41 \( 1 - 2.16e7T + 3.27e14T^{2} \)
43 \( 1 + 3.56e7T + 5.02e14T^{2} \)
47 \( 1 + 2.23e7T + 1.11e15T^{2} \)
53 \( 1 + 4.26e7T + 3.29e15T^{2} \)
59 \( 1 + 6.20e7T + 8.66e15T^{2} \)
61 \( 1 + 4.00e7T + 1.16e16T^{2} \)
67 \( 1 - 1.48e8T + 2.72e16T^{2} \)
71 \( 1 - 6.12e7T + 4.58e16T^{2} \)
73 \( 1 - 7.82e7T + 5.88e16T^{2} \)
79 \( 1 - 6.49e8T + 1.19e17T^{2} \)
83 \( 1 - 4.35e8T + 1.86e17T^{2} \)
89 \( 1 + 1.34e8T + 3.50e17T^{2} \)
97 \( 1 - 1.29e9T + 7.60e17T^{2} \)
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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

−9.511471185833838407824619156618, −9.101444094580808351492706952434, −8.217670666052634002730366354529, −6.46295358368788484593008347224, −6.08289973356855036325390809801, −4.99515659357575909072534623611, −3.55741833001007376270818328520, −2.12093537902153574039763033078, −1.14868804271285429002098073099, 0, 1.14868804271285429002098073099, 2.12093537902153574039763033078, 3.55741833001007376270818328520, 4.99515659357575909072534623611, 6.08289973356855036325390809801, 6.46295358368788484593008347224, 8.217670666052634002730366354529, 9.101444094580808351492706952434, 9.511471185833838407824619156618

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