Properties

Degree $2$
Conductor $17$
Sign $-1$
Motivic weight $9$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 26.6·2-s − 85.9·3-s + 197.·4-s − 1.46e3·5-s − 2.28e3·6-s − 446.·7-s − 8.38e3·8-s − 1.22e4·9-s − 3.88e4·10-s + 2.56e3·11-s − 1.69e4·12-s + 7.04e4·13-s − 1.18e4·14-s + 1.25e5·15-s − 3.24e5·16-s − 8.35e4·17-s − 3.27e5·18-s − 4.74e5·19-s − 2.88e5·20-s + 3.83e4·21-s + 6.83e4·22-s + 1.74e6·23-s + 7.20e5·24-s + 1.80e5·25-s + 1.87e6·26-s + 2.74e6·27-s − 8.80e4·28-s + ⋯
L(s)  = 1  + 1.17·2-s − 0.612·3-s + 0.385·4-s − 1.04·5-s − 0.721·6-s − 0.0702·7-s − 0.723·8-s − 0.624·9-s − 1.23·10-s + 0.0528·11-s − 0.236·12-s + 0.684·13-s − 0.0826·14-s + 0.640·15-s − 1.23·16-s − 0.242·17-s − 0.735·18-s − 0.834·19-s − 0.402·20-s + 0.0430·21-s + 0.0621·22-s + 1.30·23-s + 0.443·24-s + 0.0922·25-s + 0.805·26-s + 0.995·27-s − 0.0270·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(17\)
Sign: $-1$
Motivic weight: \(9\)
Character: $\chi_{17} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 17,\ (\ :9/2),\ -1)\)

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 \( 1 + 8.35e4T \)
good2 \( 1 - 26.6T + 512T^{2} \)
3 \( 1 + 85.9T + 1.96e4T^{2} \)
5 \( 1 + 1.46e3T + 1.95e6T^{2} \)
7 \( 1 + 446.T + 4.03e7T^{2} \)
11 \( 1 - 2.56e3T + 2.35e9T^{2} \)
13 \( 1 - 7.04e4T + 1.06e10T^{2} \)
19 \( 1 + 4.74e5T + 3.22e11T^{2} \)
23 \( 1 - 1.74e6T + 1.80e12T^{2} \)
29 \( 1 - 5.01e5T + 1.45e13T^{2} \)
31 \( 1 + 5.10e5T + 2.64e13T^{2} \)
37 \( 1 + 3.74e6T + 1.29e14T^{2} \)
41 \( 1 + 3.04e7T + 3.27e14T^{2} \)
43 \( 1 + 2.15e7T + 5.02e14T^{2} \)
47 \( 1 - 3.43e6T + 1.11e15T^{2} \)
53 \( 1 + 2.87e7T + 3.29e15T^{2} \)
59 \( 1 + 1.37e8T + 8.66e15T^{2} \)
61 \( 1 - 1.21e8T + 1.16e16T^{2} \)
67 \( 1 + 8.31e7T + 2.72e16T^{2} \)
71 \( 1 - 1.48e8T + 4.58e16T^{2} \)
73 \( 1 - 1.45e8T + 5.88e16T^{2} \)
79 \( 1 + 4.22e8T + 1.19e17T^{2} \)
83 \( 1 + 5.36e7T + 1.86e17T^{2} \)
89 \( 1 - 9.26e8T + 3.50e17T^{2} \)
97 \( 1 - 1.44e9T + 7.60e17T^{2} \)
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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

−15.75165868958560485058326065318, −14.72117055220996862519931652458, −13.26138271627157778527006688744, −12.01023941391589302038039713591, −11.08745444055847505602446487641, −8.601871956367533709247863674306, −6.46223949309428521033859739932, −4.91100464538083217406147490907, −3.41401003838981088558527340527, 0, 3.41401003838981088558527340527, 4.91100464538083217406147490907, 6.46223949309428521033859739932, 8.601871956367533709247863674306, 11.08745444055847505602446487641, 12.01023941391589302038039713591, 13.26138271627157778527006688744, 14.72117055220996862519931652458, 15.75165868958560485058326065318

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