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 − 4·7-s + 9-s + 4·11-s + 13-s − 17-s + 4·19-s + 4·21-s − 27-s + 2·29-s + 8·31-s − 4·33-s − 2·37-s − 39-s − 2·41-s − 8·43-s − 8·47-s + 9·49-s + 51-s + 14·53-s − 4·57-s + 12·59-s + 10·61-s − 4·63-s − 12·67-s + 8·71-s + 14·73-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.51·7-s + 1/3·9-s + 1.20·11-s + 0.277·13-s − 0.242·17-s + 0.917·19-s + 0.872·21-s − 0.192·27-s + 0.371·29-s + 1.43·31-s − 0.696·33-s − 0.328·37-s − 0.160·39-s − 0.312·41-s − 1.21·43-s − 1.16·47-s + 9/7·49-s + 0.140·51-s + 1.92·53-s − 0.529·57-s + 1.56·59-s + 1.28·61-s − 0.503·63-s − 1.46·67-s + 0.949·71-s + 1.63·73-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  =  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 \( 1 + 4 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 14 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
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

−13.00934199358873, −12.53305069609899, −11.98493864723986, −11.71524330006786, −11.40945471846391, −10.64755322725655, −10.10934089310690, −9.886009025283326, −9.458389501064155, −8.917626173209999, −8.440826142727171, −7.931397442363875, −7.066131035417094, −6.794267893133672, −6.480310126241893, −6.098360815553514, −5.287868349599625, −5.121460138966584, −4.123711427492889, −3.898338575380468, −3.306222193037355, −2.799794147105275, −2.086778955256475, −1.209730713158688, −0.8109314561488170, 0, 0.8109314561488170, 1.209730713158688, 2.086778955256475, 2.799794147105275, 3.306222193037355, 3.898338575380468, 4.123711427492889, 5.121460138966584, 5.287868349599625, 6.098360815553514, 6.480310126241893, 6.794267893133672, 7.066131035417094, 7.931397442363875, 8.440826142727171, 8.917626173209999, 9.458389501064155, 9.886009025283326, 10.10934089310690, 10.64755322725655, 11.40945471846391, 11.71524330006786, 11.98493864723986, 12.53305069609899, 13.00934199358873

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