Properties

Degree 4
Conductor $ 2^{2} \cdot 3^{2} \cdot 5^{3} \cdot 13^{2} $
Sign $-1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 1

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 3·4-s − 5-s − 8·7-s − 4·8-s + 9-s + 2·10-s + 2·13-s + 16·14-s + 5·16-s − 2·18-s − 3·20-s + 25-s − 4·26-s − 24·28-s − 12·29-s − 6·32-s + 8·35-s + 3·36-s + 4·37-s + 4·40-s − 45-s + 34·49-s − 2·50-s + 6·52-s + 32·56-s + 24·58-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s − 0.447·5-s − 3.02·7-s − 1.41·8-s + 1/3·9-s + 0.632·10-s + 0.554·13-s + 4.27·14-s + 5/4·16-s − 0.471·18-s − 0.670·20-s + 1/5·25-s − 0.784·26-s − 4.53·28-s − 2.22·29-s − 1.06·32-s + 1.35·35-s + 1/2·36-s + 0.657·37-s + 0.632·40-s − 0.149·45-s + 34/7·49-s − 0.282·50-s + 0.832·52-s + 4.27·56-s + 3.15·58-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 760500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 760500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(760500\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{3} \cdot 13^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{760500} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 760500,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5,\;13\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;5,\;13\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_1$ \( ( 1 + T )^{2} \)
3$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
5$C_1$ \( 1 + T \)
13$C_2$ \( 1 - 2 T + p T^{2} \)
good7$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
show more
show less
\[\begin{aligned} L(s) = \prod_{\mathfrak{p}\ \mathrm{bad}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s})^{-1} \prod_{\mathfrak{p}\ \mathrm{good}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s} + (N\mathfrak{p})^{-2s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−7.941231322257043910223245152113, −7.55397412652055104429359188790, −7.37431102580997813073405467010, −6.66757469343681711452650518079, −6.42217617652666865799421983967, −6.09907154602198419660987368274, −5.69930835701942778538159225157, −4.83449158660366225526812794793, −3.95694186776025806536992768009, −3.39954423361805099487890109587, −3.36585804145949552210873743484, −2.57473182165147345688800624678, −1.84915649406233336762542821373, −0.72861675057256908847367854482, 0, 0.72861675057256908847367854482, 1.84915649406233336762542821373, 2.57473182165147345688800624678, 3.36585804145949552210873743484, 3.39954423361805099487890109587, 3.95694186776025806536992768009, 4.83449158660366225526812794793, 5.69930835701942778538159225157, 6.09907154602198419660987368274, 6.42217617652666865799421983967, 6.66757469343681711452650518079, 7.37431102580997813073405467010, 7.55397412652055104429359188790, 7.941231322257043910223245152113

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