Properties

 Degree 4 Conductor $2^{2} \cdot 3^{2} \cdot 23^{3}$ Sign $1$ Motivic weight 1 Primitive no Self-dual yes Analytic rank 0

Origins of factors

Dirichlet series

 L(s)  = 1 + 4-s + 8·5-s − 3·9-s + 4·11-s − 4·13-s + 16-s − 4·17-s + 8·20-s + 23-s + 38·25-s − 3·36-s + 4·44-s − 24·45-s + 2·49-s − 4·52-s − 8·53-s + 32·55-s + 64-s − 32·65-s − 4·68-s + 12·73-s + 8·80-s + 9·81-s + 28·83-s − 32·85-s − 12·89-s + 92-s + ⋯
 L(s)  = 1 + 1/2·4-s + 3.57·5-s − 9-s + 1.20·11-s − 1.10·13-s + 1/4·16-s − 0.970·17-s + 1.78·20-s + 0.208·23-s + 38/5·25-s − 1/2·36-s + 0.603·44-s − 3.57·45-s + 2/7·49-s − 0.554·52-s − 1.09·53-s + 4.31·55-s + 1/8·64-s − 3.96·65-s − 0.485·68-s + 1.40·73-s + 0.894·80-s + 81-s + 3.07·83-s − 3.47·85-s − 1.27·89-s + 0.104·92-s + ⋯

Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 438012 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 438012 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

 $$d$$ = $$4$$ $$N$$ = $$438012$$    =    $$2^{2} \cdot 3^{2} \cdot 23^{3}$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{438012} (1, \cdot )$ Sato-Tate : $\mathrm{SU}(2)$ primitive : no self-dual : yes analytic rank = 0 Selberg data = $(4,\ 438012,\ (\ :1/2, 1/2),\ 1)$ $L(1)$ $\approx$ $4.206707818$ $L(\frac12)$ $\approx$ $4.206707818$ $L(\frac{3}{2})$ not available $L(1)$ not available

Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;3,\;23\}$, $F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4$with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;23\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_1$$\times$$C_1$ $$( 1 - T )( 1 + T )$$
3$C_2$ $$1 + p T^{2}$$
23$C_1$ $$1 - T$$
good5$C_2$ $$( 1 - 4 T + p T^{2} )^{2}$$
7$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
11$C_2$ $$( 1 - 2 T + p T^{2} )^{2}$$
13$C_2$ $$( 1 + 2 T + p T^{2} )^{2}$$
17$C_2$ $$( 1 + 2 T + p T^{2} )^{2}$$
19$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
29$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
31$C_2$ $$( 1 + p T^{2} )^{2}$$
37$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
41$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
43$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
47$C_2$ $$( 1 + p T^{2} )^{2}$$
53$C_2$ $$( 1 + 4 T + p T^{2} )^{2}$$
59$C_2$ $$( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )$$
61$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
67$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
71$C_2$ $$( 1 + p T^{2} )^{2}$$
73$C_2$ $$( 1 - 6 T + p T^{2} )^{2}$$
79$C_2$ $$( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )$$
83$C_2$ $$( 1 - 14 T + p T^{2} )^{2}$$
89$C_2$ $$( 1 + 6 T + p T^{2} )^{2}$$
97$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}