Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 4-s − 2·5-s + 4·7-s + 3·8-s − 3·9-s + 2·10-s − 2·13-s − 4·14-s − 16-s + 17-s + 3·18-s − 4·19-s + 2·20-s + 4·23-s − 25-s + 2·26-s − 4·28-s + 6·29-s + 4·31-s − 5·32-s − 34-s − 8·35-s + 3·36-s − 2·37-s + 4·38-s − 6·40-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s − 0.894·5-s + 1.51·7-s + 1.06·8-s − 9-s + 0.632·10-s − 0.554·13-s − 1.06·14-s − 1/4·16-s + 0.242·17-s + 0.707·18-s − 0.917·19-s + 0.447·20-s + 0.834·23-s − 1/5·25-s + 0.392·26-s − 0.755·28-s + 1.11·29-s + 0.718·31-s − 0.883·32-s − 0.171·34-s − 1.35·35-s + 1/2·36-s − 0.328·37-s + 0.648·38-s − 0.948·40-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(17\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{17} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 17,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.3867699383$
$L(\frac12)$  $\approx$  $0.3867699383$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 17$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p = 17$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad17 \( 1 - T \)
good2 \( 1 + T + p T^{2} \)
3 \( 1 + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−19.96219880288331, −19.01775672014589, −17.67616974075777, −16.94732940946106, −15.06935357086149, −13.97952620674689, −11.93623114442209, −10.71734099889110, −8.695686711870281, −7.819103955238084, −4.741993155413770, 4.741993155413770, 7.819103955238084, 8.695686711870281, 10.71734099889110, 11.93623114442209, 13.97952620674689, 15.06935357086149, 16.94732940946106, 17.67616974075777, 19.01775672014589, 19.96219880288331

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