# Properties

 Degree 4 Conductor $3^{2} \cdot 5^{2} \cdot 7^{2}$ Sign $1$ Motivic weight 3 Primitive no Self-dual yes Analytic rank 2

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 4·2-s + 6·3-s + 4-s − 10·5-s − 24·6-s − 14·7-s + 24·8-s + 27·9-s + 40·10-s − 92·11-s + 6·12-s + 8·13-s + 56·14-s − 60·15-s − 47·16-s − 44·17-s − 108·18-s − 108·19-s − 10·20-s − 84·21-s + 368·22-s − 320·23-s + 144·24-s + 75·25-s − 32·26-s + 108·27-s − 14·28-s + ⋯
 L(s)  = 1 − 1.41·2-s + 1.15·3-s + 1/8·4-s − 0.894·5-s − 1.63·6-s − 0.755·7-s + 1.06·8-s + 9-s + 1.26·10-s − 2.52·11-s + 0.144·12-s + 0.170·13-s + 1.06·14-s − 1.03·15-s − 0.734·16-s − 0.627·17-s − 1.41·18-s − 1.30·19-s − 0.111·20-s − 0.872·21-s + 3.56·22-s − 2.90·23-s + 1.22·24-s + 3/5·25-s − 0.241·26-s + 0.769·27-s − 0.0944·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 11025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 11025 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$11025$$    =    $$3^{2} \cdot 5^{2} \cdot 7^{2}$$ $$\varepsilon$$ = $1$ motivic weight = $$3$$ character : induced by $\chi_{105} (1, \cdot )$ primitive : no self-dual : yes analytic rank = $$2$$ Selberg data = $$(4,\ 11025,\ (\ :3/2, 3/2),\ 1)$$ $$L(2)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{3,\;5,\;7\}$,$$F_p(T)$$ is a polynomial of degree 4. If $p \in \{3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad3$C_1$ $$( 1 - p T )^{2}$$
5$C_1$ $$( 1 + p T )^{2}$$
7$C_1$ $$( 1 + p T )^{2}$$
good2$D_{4}$ $$1 + p^{2} T + 15 T^{2} + p^{5} T^{3} + p^{6} T^{4}$$
11$D_{4}$ $$1 + 92 T + 4758 T^{2} + 92 p^{3} T^{3} + p^{6} T^{4}$$
13$D_{4}$ $$1 - 8 T - 2810 T^{2} - 8 p^{3} T^{3} + p^{6} T^{4}$$
17$D_{4}$ $$1 + 44 T + 630 T^{2} + 44 p^{3} T^{3} + p^{6} T^{4}$$
19$D_{4}$ $$1 + 108 T + 13254 T^{2} + 108 p^{3} T^{3} + p^{6} T^{4}$$
23$D_{4}$ $$1 + 320 T + 47934 T^{2} + 320 p^{3} T^{3} + p^{6} T^{4}$$
29$D_{4}$ $$1 + 236 T + 61982 T^{2} + 236 p^{3} T^{3} + p^{6} T^{4}$$
31$D_{4}$ $$1 + 60 T + 8462 T^{2} + 60 p^{3} T^{3} + p^{6} T^{4}$$
37$D_{4}$ $$1 - 204 T + 108830 T^{2} - 204 p^{3} T^{3} + p^{6} T^{4}$$
41$D_{4}$ $$1 - 44 T + 106326 T^{2} - 44 p^{3} T^{3} + p^{6} T^{4}$$
43$D_{4}$ $$1 - 136 T + 81718 T^{2} - 136 p^{3} T^{3} + p^{6} T^{4}$$
47$D_{4}$ $$1 - 400 T + 106526 T^{2} - 400 p^{3} T^{3} + p^{6} T^{4}$$
53$D_{4}$ $$1 - 16 T + 260838 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4}$$
59$D_{4}$ $$1 + 464 T + 458102 T^{2} + 464 p^{3} T^{3} + p^{6} T^{4}$$
61$D_{4}$ $$1 + 684 T + 535646 T^{2} + 684 p^{3} T^{3} + p^{6} T^{4}$$
67$D_{4}$ $$1 - 736 T + 602470 T^{2} - 736 p^{3} T^{3} + p^{6} T^{4}$$
71$D_{4}$ $$1 + 740 T + 757502 T^{2} + 740 p^{3} T^{3} + p^{6} T^{4}$$
73$D_{4}$ $$1 - 424 T + 748558 T^{2} - 424 p^{3} T^{3} + p^{6} T^{4}$$
79$D_{4}$ $$1 + 408 T - 143586 T^{2} + 408 p^{3} T^{3} + p^{6} T^{4}$$
83$D_{4}$ $$1 - 608 T + 1200710 T^{2} - 608 p^{3} T^{3} + p^{6} T^{4}$$
89$D_{4}$ $$1 + 1332 T + 1790774 T^{2} + 1332 p^{3} T^{3} + p^{6} T^{4}$$
97$D_{4}$ $$1 + 2448 T + 3286542 T^{2} + 2448 p^{3} T^{3} + p^{6} T^{4}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}