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 1

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 5·7-s + 9-s − 11-s + 13-s − 17-s + 5·19-s + 5·21-s + 6·23-s − 27-s − 3·29-s + 6·31-s + 33-s − 37-s − 39-s + 7·41-s − 10·43-s + 47-s + 18·49-s + 51-s − 9·53-s − 5·57-s − 10·59-s + 4·61-s − 5·63-s − 2·67-s − 6·69-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.88·7-s + 1/3·9-s − 0.301·11-s + 0.277·13-s − 0.242·17-s + 1.14·19-s + 1.09·21-s + 1.25·23-s − 0.192·27-s − 0.557·29-s + 1.07·31-s + 0.174·33-s − 0.164·37-s − 0.160·39-s + 1.09·41-s − 1.52·43-s + 0.145·47-s + 18/7·49-s + 0.140·51-s − 1.23·53-s − 0.662·57-s − 1.30·59-s + 0.512·61-s − 0.629·63-s − 0.244·67-s − 0.722·69-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  =  1
Selberg data  =  $(2,\ 265200,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 \( 1 + 5 T + p T^{2} \)
11 \( 1 + T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 - 7 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 - T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 3 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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

−12.95200213511798, −12.70777522420172, −12.14236283449383, −11.60452215551607, −11.29574625688161, −10.64744775939357, −10.21016783669029, −9.872807443741898, −9.343415100918103, −9.041744941655396, −8.480170105542637, −7.671574974965926, −7.328598513541822, −6.854069542064535, −6.274669853642740, −6.091363704753523, −5.474866381137774, −4.891386925541318, −4.456199216321354, −3.633850200878716, −3.210816182415355, −2.924512228933767, −2.139288366696500, −1.249241253929726, −0.6733676603645918, 0, 0.6733676603645918, 1.249241253929726, 2.139288366696500, 2.924512228933767, 3.210816182415355, 3.633850200878716, 4.456199216321354, 4.891386925541318, 5.474866381137774, 6.091363704753523, 6.274669853642740, 6.854069542064535, 7.328598513541822, 7.671574974965926, 8.480170105542637, 9.041744941655396, 9.343415100918103, 9.872807443741898, 10.21016783669029, 10.64744775939357, 11.29574625688161, 11.60452215551607, 12.14236283449383, 12.70777522420172, 12.95200213511798

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