Properties

Degree 2
Conductor $ 2^{6} \cdot 3 \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  + 3-s + 4·7-s + 9-s + 2·13-s − 6·17-s + 4·19-s + 4·21-s + 27-s + 6·29-s + 8·31-s + 2·37-s + 2·39-s − 6·41-s − 4·43-s + 9·49-s − 6·51-s − 6·53-s + 4·57-s + 10·61-s + 4·63-s − 4·67-s − 2·73-s + 8·79-s + 81-s + 12·83-s + 6·87-s + 18·89-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.51·7-s + 1/3·9-s + 0.554·13-s − 1.45·17-s + 0.917·19-s + 0.872·21-s + 0.192·27-s + 1.11·29-s + 1.43·31-s + 0.328·37-s + 0.320·39-s − 0.937·41-s − 0.609·43-s + 9/7·49-s − 0.840·51-s − 0.824·53-s + 0.529·57-s + 1.28·61-s + 0.503·63-s − 0.488·67-s − 0.234·73-s + 0.900·79-s + 1/9·81-s + 1.31·83-s + 0.643·87-s + 1.90·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4800\)    =    \(2^{6} \cdot 3 \cdot 5^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{4800} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 4800,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(3.179937329\)
\(L(\frac12)\)  \(\approx\)  \(3.179937329\)
\(L(\frac{3}{2})\)   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 \( 1 \)
3 \( 1 - T \)
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

−17.75197425518445, −17.52556524440640, −16.55384957605736, −15.77744226047710, −15.39808926465555, −14.73790767408887, −14.11423526432549, −13.60258523316157, −13.17319468901469, −12.03865745983356, −11.67193574321523, −10.97200101391921, −10.38192171957206, −9.554596015735107, −8.760711413234800, −8.286230572976718, −7.802070050214001, −6.873337041746273, −6.239910348000230, −5.034218750225871, −4.692633563921229, −3.797696194546671, −2.775805121428741, −1.926770956596550, −1.028984832496703, 1.028984832496703, 1.926770956596550, 2.775805121428741, 3.797696194546671, 4.692633563921229, 5.034218750225871, 6.239910348000230, 6.873337041746273, 7.802070050214001, 8.286230572976718, 8.760711413234800, 9.554596015735107, 10.38192171957206, 10.97200101391921, 11.67193574321523, 12.03865745983356, 13.17319468901469, 13.60258523316157, 14.11423526432549, 14.73790767408887, 15.39808926465555, 15.77744226047710, 16.55384957605736, 17.52556524440640, 17.75197425518445

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