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 + 2·11-s + 13-s − 17-s + 6·19-s + 5·21-s − 4·23-s − 27-s − 9·29-s − 8·31-s − 2·33-s + 7·37-s − 39-s − 4·41-s + 43-s + 5·47-s + 18·49-s + 51-s + 4·53-s − 6·57-s + 12·59-s + 10·61-s − 5·63-s − 2·67-s + 4·69-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.88·7-s + 1/3·9-s + 0.603·11-s + 0.277·13-s − 0.242·17-s + 1.37·19-s + 1.09·21-s − 0.834·23-s − 0.192·27-s − 1.67·29-s − 1.43·31-s − 0.348·33-s + 1.15·37-s − 0.160·39-s − 0.624·41-s + 0.152·43-s + 0.729·47-s + 18/7·49-s + 0.140·51-s + 0.549·53-s − 0.794·57-s + 1.56·59-s + 1.28·61-s − 0.629·63-s − 0.244·67-s + 0.481·69-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 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 7 T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 - 5 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 - 17 T + p T^{2} \)
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

−12.94448608585277, −12.62892656687322, −12.13298600925320, −11.62790263660436, −11.28167347057371, −10.76817193071919, −10.12115546441241, −9.733660998872622, −9.508523944429266, −8.983834378961476, −8.542619671804324, −7.554930288947322, −7.408275039767322, −6.851223897827231, −6.321184682119500, −6.010861464876436, −5.422713677957196, −5.156485758068868, −4.016265532076738, −3.795768113889045, −3.525425366121184, −2.653829385119510, −2.169336905740152, −1.300182138686455, −0.6471197307303087, 0, 0.6471197307303087, 1.300182138686455, 2.169336905740152, 2.653829385119510, 3.525425366121184, 3.795768113889045, 4.016265532076738, 5.156485758068868, 5.422713677957196, 6.010861464876436, 6.321184682119500, 6.851223897827231, 7.408275039767322, 7.554930288947322, 8.542619671804324, 8.983834378961476, 9.508523944429266, 9.733660998872622, 10.12115546441241, 10.76817193071919, 11.28167347057371, 11.62790263660436, 12.13298600925320, 12.62892656687322, 12.94448608585277

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