Properties

Label 2-17e2-1.1-c9-0-26
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  − 22.4·2-s − 161.·3-s − 6.03·4-s − 1.28e3·5-s + 3.62e3·6-s − 9.35e3·7-s + 1.16e4·8-s + 6.27e3·9-s + 2.88e4·10-s − 5.71e4·11-s + 972.·12-s − 1.47e5·13-s + 2.10e5·14-s + 2.06e5·15-s − 2.59e5·16-s − 1.41e5·18-s − 4.95e5·19-s + 7.75e3·20-s + 1.50e6·21-s + 1.28e6·22-s − 2.06e6·23-s − 1.87e6·24-s − 3.02e5·25-s + 3.31e6·26-s + 2.15e6·27-s + 5.64e4·28-s − 1.09e6·29-s + ⋯
L(s)  = 1  − 0.994·2-s − 1.14·3-s − 0.0117·4-s − 0.919·5-s + 1.14·6-s − 1.47·7-s + 1.00·8-s + 0.318·9-s + 0.913·10-s − 1.17·11-s + 0.0135·12-s − 1.42·13-s + 1.46·14-s + 1.05·15-s − 0.988·16-s − 0.317·18-s − 0.872·19-s + 0.0108·20-s + 1.69·21-s + 1.16·22-s − 1.53·23-s − 1.15·24-s − 0.155·25-s + 1.42·26-s + 0.782·27-s + 0.0173·28-s − 0.287·29-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 \( 1 + 22.4T + 512T^{2} \)
3 \( 1 + 161.T + 1.96e4T^{2} \)
5 \( 1 + 1.28e3T + 1.95e6T^{2} \)
7 \( 1 + 9.35e3T + 4.03e7T^{2} \)
11 \( 1 + 5.71e4T + 2.35e9T^{2} \)
13 \( 1 + 1.47e5T + 1.06e10T^{2} \)
19 \( 1 + 4.95e5T + 3.22e11T^{2} \)
23 \( 1 + 2.06e6T + 1.80e12T^{2} \)
29 \( 1 + 1.09e6T + 1.45e13T^{2} \)
31 \( 1 + 8.20e6T + 2.64e13T^{2} \)
37 \( 1 + 1.78e7T + 1.29e14T^{2} \)
41 \( 1 - 1.40e7T + 3.27e14T^{2} \)
43 \( 1 + 2.12e6T + 5.02e14T^{2} \)
47 \( 1 - 1.57e7T + 1.11e15T^{2} \)
53 \( 1 + 3.85e7T + 3.29e15T^{2} \)
59 \( 1 - 6.56e7T + 8.66e15T^{2} \)
61 \( 1 + 4.06e7T + 1.16e16T^{2} \)
67 \( 1 - 1.28e8T + 2.72e16T^{2} \)
71 \( 1 - 1.56e8T + 4.58e16T^{2} \)
73 \( 1 + 2.48e8T + 5.88e16T^{2} \)
79 \( 1 + 5.41e8T + 1.19e17T^{2} \)
83 \( 1 - 7.51e7T + 1.86e17T^{2} \)
89 \( 1 - 4.56e8T + 3.50e17T^{2} \)
97 \( 1 - 1.00e9T + 7.60e17T^{2} \)
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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

−10.01016649542275435802107367182, −8.892240579888060816270532927520, −7.76001239807326516269980081102, −7.09737659161088507925549522805, −5.89406639033647760509667709420, −4.86830609954576653913118353969, −3.72192336977984732281818714172, −2.23327254319631461557676415891, −0.29950462214626565905603532415, 0, 0.29950462214626565905603532415, 2.23327254319631461557676415891, 3.72192336977984732281818714172, 4.86830609954576653913118353969, 5.89406639033647760509667709420, 7.09737659161088507925549522805, 7.76001239807326516269980081102, 8.892240579888060816270532927520, 10.01016649542275435802107367182

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