Properties

Degree 2
Conductor $ 3^{2} \cdot 5^{2} \cdot 23 \cdot 29 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 2

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 4-s − 3·8-s − 2·11-s − 2·13-s − 16-s − 4·17-s − 2·22-s − 23-s − 2·26-s − 29-s − 2·31-s + 5·32-s − 4·34-s − 2·37-s − 6·41-s + 4·43-s + 2·44-s − 46-s − 7·49-s + 2·52-s − 10·53-s − 58-s + 10·59-s − 2·61-s − 2·62-s + 7·64-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s − 1.06·8-s − 0.603·11-s − 0.554·13-s − 1/4·16-s − 0.970·17-s − 0.426·22-s − 0.208·23-s − 0.392·26-s − 0.185·29-s − 0.359·31-s + 0.883·32-s − 0.685·34-s − 0.328·37-s − 0.937·41-s + 0.609·43-s + 0.301·44-s − 0.147·46-s − 49-s + 0.277·52-s − 1.37·53-s − 0.131·58-s + 1.30·59-s − 0.256·61-s − 0.254·62-s + 7/8·64-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 150075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 150075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(150075\)    =    \(3^{2} \cdot 5^{2} \cdot 23 \cdot 29\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{150075} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  2
Selberg data  =  $(2,\ 150075,\ (\ :1/2),\ 1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;5,\;23,\;29\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;5,\;23,\;29\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 \)
5 \( 1 \)
23 \( 1 + T \)
29 \( 1 + T \)
good2 \( 1 - T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
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 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
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

−13.64766111188247, −13.34516365313308, −13.00286787182391, −12.43827693656069, −12.10134337479455, −11.53724125510284, −10.98298883613349, −10.56692145497327, −9.877022014011717, −9.515536787404065, −9.080152538270336, −8.361732097014120, −8.180177378768952, −7.438030731360577, −6.822646218485482, −6.460581125901733, −5.648068222891640, −5.424156575943270, −4.741723294974553, −4.417933279906166, −3.806598823055156, −3.190772690828770, −2.663332358011974, −2.055259411562859, −1.240139660554179, 0, 0, 1.240139660554179, 2.055259411562859, 2.663332358011974, 3.190772690828770, 3.806598823055156, 4.417933279906166, 4.741723294974553, 5.424156575943270, 5.648068222891640, 6.460581125901733, 6.822646218485482, 7.438030731360577, 8.180177378768952, 8.361732097014120, 9.080152538270336, 9.515536787404065, 9.877022014011717, 10.56692145497327, 10.98298883613349, 11.53724125510284, 12.10134337479455, 12.43827693656069, 13.00286787182391, 13.34516365313308, 13.64766111188247

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