Properties

Degree $2$
Conductor $17$
Sign $1$
Motivic weight $9$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 42.3·2-s + 109.·3-s + 1.28e3·4-s − 2.49e3·5-s − 4.65e3·6-s + 2.87e3·7-s − 3.27e4·8-s − 7.64e3·9-s + 1.05e5·10-s + 3.65e4·11-s + 1.41e5·12-s + 6.99e4·13-s − 1.21e5·14-s − 2.74e5·15-s + 7.32e5·16-s + 8.35e4·17-s + 3.23e5·18-s + 6.40e5·19-s − 3.21e6·20-s + 3.15e5·21-s − 1.55e6·22-s + 2.09e6·23-s − 3.59e6·24-s + 4.28e6·25-s − 2.96e6·26-s − 2.99e6·27-s + 3.69e6·28-s + ⋯
L(s)  = 1  − 1.87·2-s + 0.782·3-s + 2.51·4-s − 1.78·5-s − 1.46·6-s + 0.452·7-s − 2.83·8-s − 0.388·9-s + 3.34·10-s + 0.753·11-s + 1.96·12-s + 0.678·13-s − 0.847·14-s − 1.39·15-s + 2.79·16-s + 0.242·17-s + 0.727·18-s + 1.12·19-s − 4.48·20-s + 0.353·21-s − 1.41·22-s + 1.55·23-s − 2.21·24-s + 2.19·25-s − 1.27·26-s − 1.08·27-s + 1.13·28-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: \(0\)
Selberg data: \((2,\ 17,\ (\ :9/2),\ 1)\)

Particular Values

\(L(5)\) \(\approx\) \(0.710600\)
\(L(\frac12)\) \(\approx\) \(0.710600\)
\(L(\frac{11}{2})\) 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 \( 1 + 42.3T + 512T^{2} \)
3 \( 1 - 109.T + 1.96e4T^{2} \)
5 \( 1 + 2.49e3T + 1.95e6T^{2} \)
7 \( 1 - 2.87e3T + 4.03e7T^{2} \)
11 \( 1 - 3.65e4T + 2.35e9T^{2} \)
13 \( 1 - 6.99e4T + 1.06e10T^{2} \)
19 \( 1 - 6.40e5T + 3.22e11T^{2} \)
23 \( 1 - 2.09e6T + 1.80e12T^{2} \)
29 \( 1 - 4.99e6T + 1.45e13T^{2} \)
31 \( 1 + 5.61e6T + 2.64e13T^{2} \)
37 \( 1 - 3.47e6T + 1.29e14T^{2} \)
41 \( 1 - 4.69e5T + 3.27e14T^{2} \)
43 \( 1 - 3.50e6T + 5.02e14T^{2} \)
47 \( 1 - 1.55e6T + 1.11e15T^{2} \)
53 \( 1 - 1.03e8T + 3.29e15T^{2} \)
59 \( 1 + 7.79e7T + 8.66e15T^{2} \)
61 \( 1 - 1.79e7T + 1.16e16T^{2} \)
67 \( 1 - 8.26e7T + 2.72e16T^{2} \)
71 \( 1 + 1.03e8T + 4.58e16T^{2} \)
73 \( 1 - 1.43e7T + 5.88e16T^{2} \)
79 \( 1 - 3.90e8T + 1.19e17T^{2} \)
83 \( 1 + 3.47e8T + 1.86e17T^{2} \)
89 \( 1 + 4.95e7T + 3.50e17T^{2} \)
97 \( 1 - 6.49e8T + 7.60e17T^{2} \)
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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

−16.79051241290812732677108853013, −15.72410159685632248549507383900, −14.70391314353486816083546816534, −11.82803901583009412551996773756, −11.04999628487850933439795979740, −9.051113358499463601589838765392, −8.226234014675006777366193697748, −7.21342054371703754541035211976, −3.24490865549226043494539667760, −0.915894036795500598442491894205, 0.915894036795500598442491894205, 3.24490865549226043494539667760, 7.21342054371703754541035211976, 8.226234014675006777366193697748, 9.051113358499463601589838765392, 11.04999628487850933439795979740, 11.82803901583009412551996773756, 14.70391314353486816083546816534, 15.72410159685632248549507383900, 16.79051241290812732677108853013

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