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  − 28.1·2-s − 3.02·3-s + 281.·4-s + 762.·5-s + 85.1·6-s + 5.57e3·7-s + 6.49e3·8-s − 1.96e4·9-s − 2.14e4·10-s − 4.76e4·11-s − 850.·12-s − 9.22e4·13-s − 1.56e5·14-s − 2.30e3·15-s − 3.27e5·16-s − 8.35e4·17-s + 5.54e5·18-s − 8.37e3·19-s + 2.14e5·20-s − 1.68e4·21-s + 1.34e6·22-s + 3.64e5·23-s − 1.96e4·24-s − 1.37e6·25-s + 2.59e6·26-s + 1.19e5·27-s + 1.56e6·28-s + ⋯
L(s)  = 1  − 1.24·2-s − 0.0215·3-s + 0.549·4-s + 0.545·5-s + 0.0268·6-s + 0.877·7-s + 0.560·8-s − 0.999·9-s − 0.679·10-s − 0.981·11-s − 0.0118·12-s − 0.895·13-s − 1.09·14-s − 0.0117·15-s − 1.24·16-s − 0.242·17-s + 1.24·18-s − 0.0147·19-s + 0.299·20-s − 0.0189·21-s + 1.22·22-s + 0.271·23-s − 0.0120·24-s − 0.702·25-s + 1.11·26-s + 0.0430·27-s + 0.481·28-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 \( 1 + 28.1T + 512T^{2} \)
3 \( 1 + 3.02T + 1.96e4T^{2} \)
5 \( 1 - 762.T + 1.95e6T^{2} \)
7 \( 1 - 5.57e3T + 4.03e7T^{2} \)
11 \( 1 + 4.76e4T + 2.35e9T^{2} \)
13 \( 1 + 9.22e4T + 1.06e10T^{2} \)
19 \( 1 + 8.37e3T + 3.22e11T^{2} \)
23 \( 1 - 3.64e5T + 1.80e12T^{2} \)
29 \( 1 + 3.50e6T + 1.45e13T^{2} \)
31 \( 1 + 5.20e6T + 2.64e13T^{2} \)
37 \( 1 + 4.99e5T + 1.29e14T^{2} \)
41 \( 1 + 5.43e6T + 3.27e14T^{2} \)
43 \( 1 + 3.54e7T + 5.02e14T^{2} \)
47 \( 1 - 1.21e7T + 1.11e15T^{2} \)
53 \( 1 - 1.04e8T + 3.29e15T^{2} \)
59 \( 1 - 4.16e7T + 8.66e15T^{2} \)
61 \( 1 - 5.67e7T + 1.16e16T^{2} \)
67 \( 1 + 1.74e8T + 2.72e16T^{2} \)
71 \( 1 - 3.46e7T + 4.58e16T^{2} \)
73 \( 1 + 3.93e8T + 5.88e16T^{2} \)
79 \( 1 - 1.85e8T + 1.19e17T^{2} \)
83 \( 1 - 3.62e8T + 1.86e17T^{2} \)
89 \( 1 + 5.04e7T + 3.50e17T^{2} \)
97 \( 1 + 9.67e8T + 7.60e17T^{2} \)
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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

−16.64870546117339915250393603355, −14.83677988556092939603820121824, −13.45170435760859304360743431897, −11.41007087595098562511112276830, −10.17244007977755615560269768856, −8.787412359712218941459899605605, −7.57885072522298143691231925353, −5.24458633008911581840466429301, −2.07571285542999134835128239610, 0, 2.07571285542999134835128239610, 5.24458633008911581840466429301, 7.57885072522298143691231925353, 8.787412359712218941459899605605, 10.17244007977755615560269768856, 11.41007087595098562511112276830, 13.45170435760859304360743431897, 14.83677988556092939603820121824, 16.64870546117339915250393603355

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