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 0

Origins

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 7-s + 8-s − 6·11-s − 5·13-s − 14-s + 16-s + 6·17-s + 2·19-s − 6·22-s − 5·26-s − 28-s − 9·29-s + 5·31-s + 32-s + 6·34-s + 4·37-s + 2·38-s + 3·41-s + 4·43-s − 6·44-s + 6·47-s + 49-s − 5·52-s − 9·53-s − 56-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s − 1.80·11-s − 1.38·13-s − 0.267·14-s + 1/4·16-s + 1.45·17-s + 0.458·19-s − 1.27·22-s − 0.980·26-s − 0.188·28-s − 1.67·29-s + 0.898·31-s + 0.176·32-s + 1.02·34-s + 0.657·37-s + 0.324·38-s + 0.468·41-s + 0.609·43-s − 0.904·44-s + 0.875·47-s + 1/7·49-s − 0.693·52-s − 1.23·53-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  =  0
Selberg data  =  $(2,\ 9450,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.340323830$
$L(\frac12)$  $\approx$  $2.340323830$
$L(\frac{3}{2})$   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 \( 1 + 6 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 7 T + p T^{2} \)
67 \( 1 + 5 T + p T^{2} \)
71 \( 1 - 9 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 3 T + p T^{2} \)
89 \( 1 - 9 T + p T^{2} \)
97 \( 1 - 4 T + p T^{2} \)
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

−7.65013561627925709584199958079, −7.18390331537952058369180047085, −6.10068065166347919434075606519, −5.50231629412158597509107415557, −5.08710074644457479418458152035, −4.29648092179999106868707964293, −3.29588446114273382241926729578, −2.76733702654018778655756793057, −2.05760067697970682926821059096, −0.62383747599770320223159621579, 0.62383747599770320223159621579, 2.05760067697970682926821059096, 2.76733702654018778655756793057, 3.29588446114273382241926729578, 4.29648092179999106868707964293, 5.08710074644457479418458152035, 5.50231629412158597509107415557, 6.10068065166347919434075606519, 7.18390331537952058369180047085, 7.65013561627925709584199958079

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