Properties

Degree 2
Conductor $ 2 \cdot 7 \cdot 13^{2} $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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 + 12-s − 14-s − 3·15-s + 16-s + 6·17-s − 2·18-s + 4·19-s − 3·20-s − 21-s + 3·23-s + 24-s + 4·25-s − 5·27-s − 28-s + 6·29-s − 3·30-s + 10·31-s + 32-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 + 0.288·12-s − 0.267·14-s − 0.774·15-s + 1/4·16-s + 1.45·17-s − 0.471·18-s + 0.917·19-s − 0.670·20-s − 0.218·21-s + 0.625·23-s + 0.204·24-s + 4/5·25-s − 0.962·27-s − 0.188·28-s + 1.11·29-s − 0.547·30-s + 1.79·31-s + 0.176·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(2366\)    =    \(2 \cdot 7 \cdot 13^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{2366} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 2366,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(2.549197136\)
\(L(\frac12)\)  \(\approx\)  \(2.549197136\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

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

−8.723234695808501521366802911653, −8.116177554606699062383197527473, −7.50882438968180358884564683060, −6.74794137557946929139559683076, −5.71576798840540903108459269111, −4.93980260346123450500517401986, −3.91972626332085563597891162581, −3.26596370742597414894955978659, −2.70130097359635355412487457619, −0.909767728710544922775621814922, 0.909767728710544922775621814922, 2.70130097359635355412487457619, 3.26596370742597414894955978659, 3.91972626332085563597891162581, 4.93980260346123450500517401986, 5.71576798840540903108459269111, 6.74794137557946929139559683076, 7.50882438968180358884564683060, 8.116177554606699062383197527473, 8.723234695808501521366802911653

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