Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7^{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  − 2-s + 3-s + 4-s − 5-s − 6-s − 8-s + 9-s + 10-s − 4·11-s + 12-s − 13-s − 15-s + 16-s − 17-s − 18-s + 4·19-s − 20-s + 4·22-s − 24-s + 25-s + 26-s + 27-s − 2·29-s + 30-s − 32-s − 4·33-s + 34-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 1.20·11-s + 0.288·12-s − 0.277·13-s − 0.258·15-s + 1/4·16-s − 0.242·17-s − 0.235·18-s + 0.917·19-s − 0.223·20-s + 0.852·22-s − 0.204·24-s + 1/5·25-s + 0.196·26-s + 0.192·27-s − 0.371·29-s + 0.182·30-s − 0.176·32-s − 0.696·33-s + 0.171·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324870 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324870 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(324870\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2} \cdot 13 \cdot 17\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{324870} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 324870,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7,\;13,\;17\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;7,\;13,\;17\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 - T \)
5 \( 1 + T \)
7 \( 1 \)
13 \( 1 + T \)
17 \( 1 + T \)
good11 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−12.84958453078472, −12.31754459943642, −11.96494634463561, −11.27451996150087, −11.05396101902033, −10.51951658287088, −9.968607174586645, −9.662587883572134, −9.237906148980722, −8.591819628620595, −8.244143890714218, −7.847476306799845, −7.383925829896520, −7.085456157883486, −6.451535750791905, −5.833322373381761, −5.293676265190313, −4.805301353493589, −4.262194830004158, −3.542943506371771, −3.047402666451132, −2.694592170648061, −2.002950591835859, −1.470809671428419, −0.6518701432807705, 0, 0.6518701432807705, 1.470809671428419, 2.002950591835859, 2.694592170648061, 3.047402666451132, 3.542943506371771, 4.262194830004158, 4.805301353493589, 5.293676265190313, 5.833322373381761, 6.451535750791905, 7.085456157883486, 7.383925829896520, 7.847476306799845, 8.244143890714218, 8.591819628620595, 9.237906148980722, 9.662587883572134, 9.968607174586645, 10.51951658287088, 11.05396101902033, 11.27451996150087, 11.96494634463561, 12.31754459943642, 12.84958453078472

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