Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{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 + 4·7-s − 8-s − 2·13-s − 4·14-s + 16-s + 6·17-s − 4·19-s + 2·26-s + 4·28-s + 6·29-s + 8·31-s − 32-s − 6·34-s − 2·37-s + 4·38-s + 6·41-s + 4·43-s + 9·49-s − 2·52-s − 6·53-s − 4·56-s − 6·58-s − 10·61-s − 8·62-s + 64-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 1.51·7-s − 0.353·8-s − 0.554·13-s − 1.06·14-s + 1/4·16-s + 1.45·17-s − 0.917·19-s + 0.392·26-s + 0.755·28-s + 1.11·29-s + 1.43·31-s − 0.176·32-s − 1.02·34-s − 0.328·37-s + 0.648·38-s + 0.937·41-s + 0.609·43-s + 9/7·49-s − 0.277·52-s − 0.824·53-s − 0.534·56-s − 0.787·58-s − 1.28·61-s − 1.01·62-s + 1/8·64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{450} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 450,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(1.19638\)
\(L(\frac12)\)  \(\approx\)  \(1.19638\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 - 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
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 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
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

−10.91744252348428399819557462541, −10.27409415171032704331958668230, −9.266851904332541042335427380588, −8.128521468164053694071078641328, −7.84831740970949093070141074865, −6.58767052470959898247985953468, −5.37349128799592068356886197367, −4.36655234059998826349058539516, −2.65454187476862090428654397793, −1.28283795797463795719739258116, 1.28283795797463795719739258116, 2.65454187476862090428654397793, 4.36655234059998826349058539516, 5.37349128799592068356886197367, 6.58767052470959898247985953468, 7.84831740970949093070141074865, 8.128521468164053694071078641328, 9.266851904332541042335427380588, 10.27409415171032704331958668230, 10.91744252348428399819557462541

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