Properties

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

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 11.8·2-s − 67.6·3-s − 372.·4-s + 2.39e3·5-s − 799.·6-s − 1.13e4·7-s − 1.04e4·8-s − 1.51e4·9-s + 2.82e4·10-s − 1.77e4·11-s + 2.51e4·12-s − 7.61e4·13-s − 1.34e5·14-s − 1.61e5·15-s + 6.70e4·16-s − 8.35e4·17-s − 1.78e5·18-s + 6.61e5·19-s − 8.89e5·20-s + 7.68e5·21-s − 2.09e5·22-s − 1.64e6·23-s + 7.07e5·24-s + 3.76e6·25-s − 8.99e5·26-s + 2.35e6·27-s + 4.22e6·28-s + ⋯
L(s)  = 1  + 0.522·2-s − 0.482·3-s − 0.727·4-s + 1.71·5-s − 0.251·6-s − 1.78·7-s − 0.902·8-s − 0.767·9-s + 0.893·10-s − 0.365·11-s + 0.350·12-s − 0.739·13-s − 0.933·14-s − 0.825·15-s + 0.255·16-s − 0.242·17-s − 0.400·18-s + 1.16·19-s − 1.24·20-s + 0.862·21-s − 0.190·22-s − 1.22·23-s + 0.435·24-s + 1.92·25-s − 0.386·26-s + 0.852·27-s + 1.29·28-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 \( 1 - 11.8T + 512T^{2} \)
3 \( 1 + 67.6T + 1.96e4T^{2} \)
5 \( 1 - 2.39e3T + 1.95e6T^{2} \)
7 \( 1 + 1.13e4T + 4.03e7T^{2} \)
11 \( 1 + 1.77e4T + 2.35e9T^{2} \)
13 \( 1 + 7.61e4T + 1.06e10T^{2} \)
19 \( 1 - 6.61e5T + 3.22e11T^{2} \)
23 \( 1 + 1.64e6T + 1.80e12T^{2} \)
29 \( 1 - 1.49e6T + 1.45e13T^{2} \)
31 \( 1 + 4.40e6T + 2.64e13T^{2} \)
37 \( 1 + 5.62e6T + 1.29e14T^{2} \)
41 \( 1 - 2.29e7T + 3.27e14T^{2} \)
43 \( 1 + 7.92e6T + 5.02e14T^{2} \)
47 \( 1 + 5.69e7T + 1.11e15T^{2} \)
53 \( 1 - 2.56e6T + 3.29e15T^{2} \)
59 \( 1 + 6.87e7T + 8.66e15T^{2} \)
61 \( 1 + 1.09e8T + 1.16e16T^{2} \)
67 \( 1 - 1.44e8T + 2.72e16T^{2} \)
71 \( 1 + 6.07e7T + 4.58e16T^{2} \)
73 \( 1 + 1.68e8T + 5.88e16T^{2} \)
79 \( 1 - 1.00e8T + 1.19e17T^{2} \)
83 \( 1 + 5.82e8T + 1.86e17T^{2} \)
89 \( 1 + 1.56e7T + 3.50e17T^{2} \)
97 \( 1 - 1.64e9T + 7.60e17T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−16.26656602288058166260962394319, −14.26333476982777904808795113512, −13.38911912367261869046684507638, −12.40350042918272872180417539187, −10.03847427370142506228505963970, −9.293119528811000527869424931187, −6.26986494640548247187786581752, −5.40054711480333106571365523165, −2.92077445352434282481516541031, 0, 2.92077445352434282481516541031, 5.40054711480333106571365523165, 6.26986494640548247187786581752, 9.293119528811000527869424931187, 10.03847427370142506228505963970, 12.40350042918272872180417539187, 13.38911912367261869046684507638, 14.26333476982777904808795113512, 16.26656602288058166260962394319

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