Properties

Label 2-17e2-1.1-c9-0-193
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  + 38.8·2-s + 39.5·3-s + 1.00e3·4-s + 1.32e3·5-s + 1.53e3·6-s − 2.03e3·7-s + 1.89e4·8-s − 1.81e4·9-s + 5.15e4·10-s − 7.97e4·11-s + 3.95e4·12-s + 1.98e4·13-s − 7.89e4·14-s + 5.25e4·15-s + 2.25e5·16-s − 7.04e5·18-s − 4.30e5·19-s + 1.32e6·20-s − 8.04e4·21-s − 3.10e6·22-s − 1.47e6·23-s + 7.51e5·24-s − 1.93e5·25-s + 7.72e5·26-s − 1.49e6·27-s − 2.03e6·28-s + 5.01e6·29-s + ⋯
L(s)  = 1  + 1.71·2-s + 0.282·3-s + 1.95·4-s + 0.949·5-s + 0.485·6-s − 0.319·7-s + 1.63·8-s − 0.920·9-s + 1.63·10-s − 1.64·11-s + 0.551·12-s + 0.192·13-s − 0.549·14-s + 0.267·15-s + 0.861·16-s − 1.58·18-s − 0.757·19-s + 1.85·20-s − 0.0902·21-s − 2.82·22-s − 1.10·23-s + 0.462·24-s − 0.0989·25-s + 0.331·26-s − 0.542·27-s − 0.624·28-s + 1.31·29-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 \( 1 - 38.8T + 512T^{2} \)
3 \( 1 - 39.5T + 1.96e4T^{2} \)
5 \( 1 - 1.32e3T + 1.95e6T^{2} \)
7 \( 1 + 2.03e3T + 4.03e7T^{2} \)
11 \( 1 + 7.97e4T + 2.35e9T^{2} \)
13 \( 1 - 1.98e4T + 1.06e10T^{2} \)
19 \( 1 + 4.30e5T + 3.22e11T^{2} \)
23 \( 1 + 1.47e6T + 1.80e12T^{2} \)
29 \( 1 - 5.01e6T + 1.45e13T^{2} \)
31 \( 1 + 7.99e6T + 2.64e13T^{2} \)
37 \( 1 + 6.88e5T + 1.29e14T^{2} \)
41 \( 1 + 8.23e6T + 3.27e14T^{2} \)
43 \( 1 - 4.37e7T + 5.02e14T^{2} \)
47 \( 1 + 3.64e7T + 1.11e15T^{2} \)
53 \( 1 - 5.12e7T + 3.29e15T^{2} \)
59 \( 1 - 1.34e8T + 8.66e15T^{2} \)
61 \( 1 + 9.39e7T + 1.16e16T^{2} \)
67 \( 1 - 1.29e8T + 2.72e16T^{2} \)
71 \( 1 + 2.39e7T + 4.58e16T^{2} \)
73 \( 1 + 4.14e8T + 5.88e16T^{2} \)
79 \( 1 + 3.87e8T + 1.19e17T^{2} \)
83 \( 1 - 2.80e8T + 1.86e17T^{2} \)
89 \( 1 - 9.07e8T + 3.50e17T^{2} \)
97 \( 1 - 1.62e9T + 7.60e17T^{2} \)
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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

−10.07283113546335625042861436191, −8.724735364675818805572695137688, −7.58519879645713616531275698006, −6.23883018743904924456708208794, −5.73530956695089997663572501891, −4.89191036796597717980712584494, −3.61633832007923561358641848009, −2.61255341896204621002751537693, −2.07309927362883809403528651159, 0, 2.07309927362883809403528651159, 2.61255341896204621002751537693, 3.61633832007923561358641848009, 4.89191036796597717980712584494, 5.73530956695089997663572501891, 6.23883018743904924456708208794, 7.58519879645713616531275698006, 8.724735364675818805572695137688, 10.07283113546335625042861436191

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