Properties

Degree 2
Conductor $ 2^{3} \cdot 5 \cdot 11^{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  + 5-s + 4·7-s − 3·9-s + 2·13-s − 2·17-s − 4·19-s + 4·23-s + 25-s + 2·29-s − 8·31-s + 4·35-s + 6·37-s + 6·41-s + 8·43-s − 3·45-s + 4·47-s + 9·49-s + 6·53-s − 4·59-s + 2·61-s − 12·63-s + 2·65-s + 8·67-s + 6·73-s + 9·81-s + 16·83-s − 2·85-s + ⋯
L(s)  = 1  + 0.447·5-s + 1.51·7-s − 9-s + 0.554·13-s − 0.485·17-s − 0.917·19-s + 0.834·23-s + 1/5·25-s + 0.371·29-s − 1.43·31-s + 0.676·35-s + 0.986·37-s + 0.937·41-s + 1.21·43-s − 0.447·45-s + 0.583·47-s + 9/7·49-s + 0.824·53-s − 0.520·59-s + 0.256·61-s − 1.51·63-s + 0.248·65-s + 0.977·67-s + 0.702·73-s + 81-s + 1.75·83-s − 0.216·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4840\)    =    \(2^{3} \cdot 5 \cdot 11^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{4840} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 4840,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.434892033$
$L(\frac12)$  $\approx$  $2.434892033$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;5,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 - T \)
11 \( 1 \)
good3 \( 1 + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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

−17.92971335695733, −17.17492442690531, −16.81998100513025, −16.03501709133251, −15.08637612017040, −14.77765510757337, −14.17498784610368, −13.66784632466129, −12.89880973012015, −12.25643471137664, −11.33367087396289, −10.98399032722794, −10.66087840225076, −9.402313447498214, −8.903529359998923, −8.319898091581097, −7.703039582231378, −6.816448236659735, −5.940284449975691, −5.418106824457115, −4.624870519015605, −3.871380955868794, −2.642645276424593, −1.998734266789385, −0.8803150593745565, 0.8803150593745565, 1.998734266789385, 2.642645276424593, 3.871380955868794, 4.624870519015605, 5.418106824457115, 5.940284449975691, 6.816448236659735, 7.703039582231378, 8.319898091581097, 8.903529359998923, 9.402313447498214, 10.66087840225076, 10.98399032722794, 11.33367087396289, 12.25643471137664, 12.89880973012015, 13.66784632466129, 14.17498784610368, 14.77765510757337, 15.08637612017040, 16.03501709133251, 16.81998100513025, 17.17492442690531, 17.92971335695733

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