Properties

Degree 2
Conductor $ 2 \cdot 13 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s − 3·5-s − 6-s − 7-s − 8-s − 2·9-s + 3·10-s + 6·11-s + 12-s + 13-s + 14-s − 3·15-s + 16-s − 3·17-s + 2·18-s + 2·19-s − 3·20-s − 21-s − 6·22-s − 24-s + 4·25-s − 26-s − 5·27-s − 28-s + 6·29-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.34·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s − 2/3·9-s + 0.948·10-s + 1.80·11-s + 0.288·12-s + 0.277·13-s + 0.267·14-s − 0.774·15-s + 1/4·16-s − 0.727·17-s + 0.471·18-s + 0.458·19-s − 0.670·20-s − 0.218·21-s − 1.27·22-s − 0.204·24-s + 4/5·25-s − 0.196·26-s − 0.962·27-s − 0.188·28-s + 1.11·29-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(26\)    =    \(2 \cdot 13\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{26} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 26,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.5155766512$
$L(\frac12)$  $\approx$  $0.5155766512$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;13\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;13\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + T \)
13 \( 1 - T \)
good3 \( 1 - T + p T^{2} \)
5 \( 1 + 3 T + p T^{2} \)
7 \( 1 + T + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 + 3 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
show more
show less
\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−19.69339975173152, −19.14446932223225, −17.51877200404443, −16.32893427071431, −15.26212233610873, −14.10812305484995, −12.10810868275674, −11.21486999262722, −9.252086119982470, −8.289125051595985, −6.764231323278578, −3.640287616260134, 3.640287616260134, 6.764231323278578, 8.289125051595985, 9.252086119982470, 11.21486999262722, 12.10810868275674, 14.10812305484995, 15.26212233610873, 16.32893427071431, 17.51877200404443, 19.14446932223225, 19.69339975173152

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