Properties

Degree 2
Conductor $ 3^{2} \cdot 5^{2} \cdot 7 \cdot 11^{2} $
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 − 4-s + 7-s + 3·8-s − 6·13-s − 14-s − 16-s − 2·17-s + 8·19-s + 8·23-s + 6·26-s − 28-s − 2·29-s + 4·31-s − 5·32-s + 2·34-s + 2·37-s − 8·38-s − 6·41-s + 4·43-s − 8·46-s + 8·47-s + 49-s + 6·52-s + 10·53-s + 3·56-s + 2·58-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s + 0.377·7-s + 1.06·8-s − 1.66·13-s − 0.267·14-s − 1/4·16-s − 0.485·17-s + 1.83·19-s + 1.66·23-s + 1.17·26-s − 0.188·28-s − 0.371·29-s + 0.718·31-s − 0.883·32-s + 0.342·34-s + 0.328·37-s − 1.29·38-s − 0.937·41-s + 0.609·43-s − 1.17·46-s + 1.16·47-s + 1/7·49-s + 0.832·52-s + 1.37·53-s + 0.400·56-s + 0.262·58-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 190575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\n\]
\[\begin{aligned} \Lambda(s)=\mathstrut & 190575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\n\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(190575\)    =    \(3^{2} \cdot 5^{2} \cdot 7 \cdot 11^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{190575} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 190575,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.825876929$
$L(\frac12)$  $\approx$  $1.825876929$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;5,\;7,\;11\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;5,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 - T \)
11 \( 1 \)
good2 \( 1 + T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
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 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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

−13.17568306281396, −12.60347097187665, −12.11934049255719, −11.68172158496988, −11.16513003250985, −10.68130738499555, −10.10239652995665, −9.762579051167062, −9.347154166718964, −8.845571707275328, −8.525456746643527, −7.713915194847941, −7.397297643653930, −7.202728143460093, −6.501988462912632, −5.521940394139265, −5.311418024188911, −4.696526732880632, −4.448675590202783, −3.585424404118599, −2.994499379531987, −2.387080795046817, −1.734824362380236, −0.8725433982422618, −0.5881733644808502, 0.5881733644808502, 0.8725433982422618, 1.734824362380236, 2.387080795046817, 2.994499379531987, 3.585424404118599, 4.448675590202783, 4.696526732880632, 5.311418024188911, 5.521940394139265, 6.501988462912632, 7.202728143460093, 7.397297643653930, 7.713915194847941, 8.525456746643527, 8.845571707275328, 9.347154166718964, 9.762579051167062, 10.10239652995665, 10.68130738499555, 11.16513003250985, 11.68172158496988, 12.11934049255719, 12.60347097187665, 13.17568306281396

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