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  − 42.0·2-s − 256.·3-s + 1.25e3·4-s + 1.40e3·5-s + 1.08e4·6-s − 5.13e3·7-s − 3.13e4·8-s + 4.62e4·9-s − 5.91e4·10-s + 2.65e4·11-s − 3.22e5·12-s + 7.14e4·13-s + 2.16e5·14-s − 3.61e5·15-s + 6.74e5·16-s − 8.35e4·17-s − 1.94e6·18-s − 5.48e5·19-s + 1.76e6·20-s + 1.31e6·21-s − 1.11e6·22-s + 1.15e6·23-s + 8.04e6·24-s + 2.72e4·25-s − 3.00e6·26-s − 6.82e6·27-s − 6.46e6·28-s + ⋯
L(s)  = 1  − 1.85·2-s − 1.83·3-s + 2.45·4-s + 1.00·5-s + 3.40·6-s − 0.808·7-s − 2.70·8-s + 2.34·9-s − 1.87·10-s + 0.546·11-s − 4.49·12-s + 0.693·13-s + 1.50·14-s − 1.84·15-s + 2.57·16-s − 0.242·17-s − 4.36·18-s − 0.965·19-s + 2.47·20-s + 1.48·21-s − 1.01·22-s + 0.864·23-s + 4.95·24-s + 0.0139·25-s − 1.28·26-s − 2.47·27-s − 1.98·28-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 \( 1 + 42.0T + 512T^{2} \)
3 \( 1 + 256.T + 1.96e4T^{2} \)
5 \( 1 - 1.40e3T + 1.95e6T^{2} \)
7 \( 1 + 5.13e3T + 4.03e7T^{2} \)
11 \( 1 - 2.65e4T + 2.35e9T^{2} \)
13 \( 1 - 7.14e4T + 1.06e10T^{2} \)
19 \( 1 + 5.48e5T + 3.22e11T^{2} \)
23 \( 1 - 1.15e6T + 1.80e12T^{2} \)
29 \( 1 - 1.44e6T + 1.45e13T^{2} \)
31 \( 1 + 6.05e6T + 2.64e13T^{2} \)
37 \( 1 + 9.50e6T + 1.29e14T^{2} \)
41 \( 1 - 1.75e7T + 3.27e14T^{2} \)
43 \( 1 + 2.06e7T + 5.02e14T^{2} \)
47 \( 1 - 3.15e7T + 1.11e15T^{2} \)
53 \( 1 + 1.02e8T + 3.29e15T^{2} \)
59 \( 1 + 5.95e7T + 8.66e15T^{2} \)
61 \( 1 - 5.34e7T + 1.16e16T^{2} \)
67 \( 1 + 6.45e7T + 2.72e16T^{2} \)
71 \( 1 + 2.71e8T + 4.58e16T^{2} \)
73 \( 1 - 2.75e8T + 5.88e16T^{2} \)
79 \( 1 + 4.33e8T + 1.19e17T^{2} \)
83 \( 1 - 1.23e8T + 1.86e17T^{2} \)
89 \( 1 + 8.87e8T + 3.50e17T^{2} \)
97 \( 1 + 4.41e8T + 7.60e17T^{2} \)
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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

−16.74617914817934616029540958849, −15.74625118886963632112456694456, −12.72126114348821388618046082561, −11.23240595320376213511309813856, −10.34105374452628142326963725185, −9.231449938266936398061284756306, −6.80615706385180649153702521673, −6.01247830173674070548198063867, −1.46726334419730167440098000473, 0, 1.46726334419730167440098000473, 6.01247830173674070548198063867, 6.80615706385180649153702521673, 9.231449938266936398061284756306, 10.34105374452628142326963725185, 11.23240595320376213511309813856, 12.72126114348821388618046082561, 15.74625118886963632112456694456, 16.74617914817934616029540958849

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