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 − 2·13-s + 14-s + 16-s − 8·17-s + 4·19-s − 4·23-s − 2·26-s + 28-s − 7·29-s + 31-s + 32-s − 8·34-s + 7·37-s + 4·38-s + 9·41-s + 2·43-s − 4·46-s − 7·47-s + 49-s − 2·52-s − 6·53-s + 56-s − 7·58-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s − 0.554·13-s + 0.267·14-s + 1/4·16-s − 1.94·17-s + 0.917·19-s − 0.834·23-s − 0.392·26-s + 0.188·28-s − 1.29·29-s + 0.179·31-s + 0.176·32-s − 1.37·34-s + 1.15·37-s + 0.648·38-s + 1.40·41-s + 0.304·43-s − 0.589·46-s − 1.02·47-s + 1/7·49-s − 0.277·52-s − 0.824·53-s + 0.133·56-s − 0.919·58-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 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 8 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 7 T + p T^{2} \)
31 \( 1 - T + p T^{2} \)
37 \( 1 - 7 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + 7 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + T + p T^{2} \)
61 \( 1 - 3 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 3 T + p T^{2} \)
73 \( 1 + 13 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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

−7.34796503845527995756522262588, −6.59971939959905125487482710903, −5.91298588760313812273303022344, −5.28532016776941831634982634580, −4.38906867166379391911397291572, −4.14666907065908716697284186801, −2.97502163688704776954663339164, −2.32638302609667582175952259427, −1.47668878881998672802258584317, 0, 1.47668878881998672802258584317, 2.32638302609667582175952259427, 2.97502163688704776954663339164, 4.14666907065908716697284186801, 4.38906867166379391911397291572, 5.28532016776941831634982634580, 5.91298588760313812273303022344, 6.59971939959905125487482710903, 7.34796503845527995756522262588

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