Properties

Label 2-17e2-1.1-c9-0-157
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  − 20.0·2-s + 107.·3-s − 109.·4-s + 2.67e3·5-s − 2.14e3·6-s − 1.98e3·7-s + 1.24e4·8-s − 8.23e3·9-s − 5.36e4·10-s − 3.76e4·11-s − 1.17e4·12-s − 2.51e4·13-s + 3.97e4·14-s + 2.86e5·15-s − 1.93e5·16-s + 1.65e5·18-s + 6.23e5·19-s − 2.93e5·20-s − 2.12e5·21-s + 7.55e5·22-s + 8.89e4·23-s + 1.33e6·24-s + 5.20e6·25-s + 5.04e5·26-s − 2.98e6·27-s + 2.17e5·28-s − 1.16e5·29-s + ⋯
L(s)  = 1  − 0.886·2-s + 0.762·3-s − 0.214·4-s + 1.91·5-s − 0.675·6-s − 0.312·7-s + 1.07·8-s − 0.418·9-s − 1.69·10-s − 0.775·11-s − 0.163·12-s − 0.244·13-s + 0.276·14-s + 1.46·15-s − 0.739·16-s + 0.370·18-s + 1.09·19-s − 0.410·20-s − 0.237·21-s + 0.687·22-s + 0.0662·23-s + 0.820·24-s + 2.66·25-s + 0.216·26-s − 1.08·27-s + 0.0669·28-s − 0.0307·29-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 \( 1 + 20.0T + 512T^{2} \)
3 \( 1 - 107.T + 1.96e4T^{2} \)
5 \( 1 - 2.67e3T + 1.95e6T^{2} \)
7 \( 1 + 1.98e3T + 4.03e7T^{2} \)
11 \( 1 + 3.76e4T + 2.35e9T^{2} \)
13 \( 1 + 2.51e4T + 1.06e10T^{2} \)
19 \( 1 - 6.23e5T + 3.22e11T^{2} \)
23 \( 1 - 8.89e4T + 1.80e12T^{2} \)
29 \( 1 + 1.16e5T + 1.45e13T^{2} \)
31 \( 1 + 8.85e6T + 2.64e13T^{2} \)
37 \( 1 - 1.75e7T + 1.29e14T^{2} \)
41 \( 1 + 2.89e7T + 3.27e14T^{2} \)
43 \( 1 + 4.22e6T + 5.02e14T^{2} \)
47 \( 1 + 3.58e7T + 1.11e15T^{2} \)
53 \( 1 + 1.03e7T + 3.29e15T^{2} \)
59 \( 1 - 6.28e7T + 8.66e15T^{2} \)
61 \( 1 - 1.06e8T + 1.16e16T^{2} \)
67 \( 1 + 9.64e7T + 2.72e16T^{2} \)
71 \( 1 + 2.78e8T + 4.58e16T^{2} \)
73 \( 1 - 3.95e8T + 5.88e16T^{2} \)
79 \( 1 + 5.96e8T + 1.19e17T^{2} \)
83 \( 1 + 7.90e8T + 1.86e17T^{2} \)
89 \( 1 - 8.30e8T + 3.50e17T^{2} \)
97 \( 1 + 2.87e8T + 7.60e17T^{2} \)
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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

−9.674046211200899912838755824447, −9.026228688249168825977124908230, −8.161221727597073361857587163313, −7.08359883667994441463285748949, −5.75685503716694428887489272707, −5.02341081628304762758389773585, −3.21350763898690252330665859537, −2.24360590379153329186962227147, −1.37564468581673252429053325234, 0, 1.37564468581673252429053325234, 2.24360590379153329186962227147, 3.21350763898690252330665859537, 5.02341081628304762758389773585, 5.75685503716694428887489272707, 7.08359883667994441463285748949, 8.161221727597073361857587163313, 9.026228688249168825977124908230, 9.674046211200899912838755824447

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