Properties

Degree 2
Conductor $ 2 \cdot 3^{3} \cdot 5^{2} \cdot 7 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 7-s + 8-s − 3·11-s + 13-s − 14-s + 16-s − 19-s − 3·22-s + 6·23-s + 26-s − 28-s − 6·29-s − 4·31-s + 32-s − 2·37-s − 38-s − 3·41-s + 43-s − 3·44-s + 6·46-s + 9·47-s + 49-s + 52-s − 3·53-s − 56-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s − 0.904·11-s + 0.277·13-s − 0.267·14-s + 1/4·16-s − 0.229·19-s − 0.639·22-s + 1.25·23-s + 0.196·26-s − 0.188·28-s − 1.11·29-s − 0.718·31-s + 0.176·32-s − 0.328·37-s − 0.162·38-s − 0.468·41-s + 0.152·43-s − 0.452·44-s + 0.884·46-s + 1.31·47-s + 1/7·49-s + 0.138·52-s − 0.412·53-s − 0.133·56-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(9450\)    =    \(2 \cdot 3^{3} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{9450} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 9450,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5,\;7\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;7\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 - T \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 + T \)
good11 \( 1 + 3 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 + 11 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 3 T + p T^{2} \)
89 \( 1 - 9 T + p T^{2} \)
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

−7.33525161574670180844916310001, −6.62142311322130506307084822210, −5.84926691606024658291065787758, −5.33086774531301444160004851915, −4.62539689715339366614752787335, −3.78645006818420864823702282016, −3.10859672921246680827878349694, −2.39481983228610393809744814666, −1.39030009645632472393134527490, 0, 1.39030009645632472393134527490, 2.39481983228610393809744814666, 3.10859672921246680827878349694, 3.78645006818420864823702282016, 4.62539689715339366614752787335, 5.33086774531301444160004851915, 5.84926691606024658291065787758, 6.62142311322130506307084822210, 7.33525161574670180844916310001

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