Properties

Label 2-17e2-1.1-c9-0-128
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  − 35.4·2-s + 64.4·3-s + 747.·4-s + 1.77e3·5-s − 2.28e3·6-s − 1.60e3·7-s − 8.36e3·8-s − 1.55e4·9-s − 6.31e4·10-s − 3.81e4·11-s + 4.81e4·12-s − 1.16e5·13-s + 5.68e4·14-s + 1.14e5·15-s − 8.59e4·16-s + 5.51e5·18-s + 8.50e5·19-s + 1.33e6·20-s − 1.03e5·21-s + 1.35e6·22-s + 1.05e6·23-s − 5.39e5·24-s + 1.21e6·25-s + 4.13e6·26-s − 2.26e6·27-s − 1.19e6·28-s − 1.44e6·29-s + ⋯
L(s)  = 1  − 1.56·2-s + 0.459·3-s + 1.46·4-s + 1.27·5-s − 0.720·6-s − 0.252·7-s − 0.722·8-s − 0.789·9-s − 1.99·10-s − 0.786·11-s + 0.670·12-s − 1.13·13-s + 0.395·14-s + 0.584·15-s − 0.327·16-s + 1.23·18-s + 1.49·19-s + 1.85·20-s − 0.115·21-s + 1.23·22-s + 0.785·23-s − 0.331·24-s + 0.621·25-s + 1.77·26-s − 0.821·27-s − 0.368·28-s − 0.379·29-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 \( 1 + 35.4T + 512T^{2} \)
3 \( 1 - 64.4T + 1.96e4T^{2} \)
5 \( 1 - 1.77e3T + 1.95e6T^{2} \)
7 \( 1 + 1.60e3T + 4.03e7T^{2} \)
11 \( 1 + 3.81e4T + 2.35e9T^{2} \)
13 \( 1 + 1.16e5T + 1.06e10T^{2} \)
19 \( 1 - 8.50e5T + 3.22e11T^{2} \)
23 \( 1 - 1.05e6T + 1.80e12T^{2} \)
29 \( 1 + 1.44e6T + 1.45e13T^{2} \)
31 \( 1 - 5.24e6T + 2.64e13T^{2} \)
37 \( 1 + 5.51e6T + 1.29e14T^{2} \)
41 \( 1 - 2.30e7T + 3.27e14T^{2} \)
43 \( 1 - 1.73e7T + 5.02e14T^{2} \)
47 \( 1 - 5.62e7T + 1.11e15T^{2} \)
53 \( 1 + 7.72e7T + 3.29e15T^{2} \)
59 \( 1 - 9.35e7T + 8.66e15T^{2} \)
61 \( 1 + 1.85e8T + 1.16e16T^{2} \)
67 \( 1 + 2.00e8T + 2.72e16T^{2} \)
71 \( 1 + 7.05e7T + 4.58e16T^{2} \)
73 \( 1 - 2.21e8T + 5.88e16T^{2} \)
79 \( 1 - 6.34e8T + 1.19e17T^{2} \)
83 \( 1 + 5.88e7T + 1.86e17T^{2} \)
89 \( 1 + 7.76e8T + 3.50e17T^{2} \)
97 \( 1 - 6.87e8T + 7.60e17T^{2} \)
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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

−9.472290252264847720092687417527, −9.183964225536682292004552780925, −7.960448740567222999113063012794, −7.28640441742441167407221632659, −6.02648181652324292586434668519, −5.05100269671711646739478597376, −2.87488070142401114884026295175, −2.33559265205905842456688350955, −1.13631391351414857451093452455, 0, 1.13631391351414857451093452455, 2.33559265205905842456688350955, 2.87488070142401114884026295175, 5.05100269671711646739478597376, 6.02648181652324292586434668519, 7.28640441742441167407221632659, 7.960448740567222999113063012794, 9.183964225536682292004552780925, 9.472290252264847720092687417527

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