Properties

Degree 2
Conductor $ 2^{4} \cdot 3 \cdot 5^{2} \cdot 13 \cdot 17 $
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 − 2·7-s + 9-s − 4·11-s − 13-s + 17-s + 2·21-s + 8·23-s − 27-s + 6·29-s + 2·31-s + 4·33-s − 4·37-s + 39-s − 2·41-s − 4·43-s + 2·47-s − 3·49-s − 51-s − 10·53-s − 10·59-s + 14·61-s − 2·63-s − 12·67-s − 8·69-s + 8·71-s − 8·73-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.755·7-s + 1/3·9-s − 1.20·11-s − 0.277·13-s + 0.242·17-s + 0.436·21-s + 1.66·23-s − 0.192·27-s + 1.11·29-s + 0.359·31-s + 0.696·33-s − 0.657·37-s + 0.160·39-s − 0.312·41-s − 0.609·43-s + 0.291·47-s − 3/7·49-s − 0.140·51-s − 1.37·53-s − 1.30·59-s + 1.79·61-s − 0.251·63-s − 1.46·67-s − 0.963·69-s + 0.949·71-s − 0.936·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 265200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 265200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(265200\)    =    \(2^{4} \cdot 3 \cdot 5^{2} \cdot 13 \cdot 17\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{265200} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 265200,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.8963489742$
$L(\frac12)$  $\approx$  $0.8963489742$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;13,\;17\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;13,\;17\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
5 \( 1 \)
13 \( 1 + T \)
17 \( 1 - T \)
good7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
89 \( 1 + 8 T + p T^{2} \)
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

−12.83943125602693, −12.38177674350631, −11.90274011700430, −11.46720750689241, −10.78365014950355, −10.57159956443638, −10.13350944435999, −9.607605136294510, −9.209234278188286, −8.593906270201475, −8.072047258028582, −7.641140324777726, −7.016528205176217, −6.658781658002780, −6.252799610835457, −5.536395729113923, −5.158575912135742, −4.763906024741018, −4.219992283273907, −3.253282051624654, −3.114817515398658, −2.509927578436376, −1.713448200978329, −0.9982202364369472, −0.3037546373351609, 0.3037546373351609, 0.9982202364369472, 1.713448200978329, 2.509927578436376, 3.114817515398658, 3.253282051624654, 4.219992283273907, 4.763906024741018, 5.158575912135742, 5.536395729113923, 6.252799610835457, 6.658781658002780, 7.016528205176217, 7.641140324777726, 8.072047258028582, 8.593906270201475, 9.209234278188286, 9.607605136294510, 10.13350944435999, 10.57159956443638, 10.78365014950355, 11.46720750689241, 11.90274011700430, 12.38177674350631, 12.83943125602693

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