# Properties

 Degree 4 Conductor $3^{2} \cdot 5^{2}$ Sign $1$ Motivic weight 7 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 7·2-s + 54·3-s − 69·4-s + 250·5-s + 378·6-s + 1.30e3·7-s − 413·8-s + 2.18e3·9-s + 1.75e3·10-s + 3.44e3·11-s − 3.72e3·12-s − 8.98e3·13-s + 9.12e3·14-s + 1.35e4·15-s − 4.86e3·16-s − 5.49e3·17-s + 1.53e4·18-s − 4.95e4·19-s − 1.72e4·20-s + 7.04e4·21-s + 2.41e4·22-s + 9.18e4·23-s − 2.23e4·24-s + 4.68e4·25-s − 6.29e4·26-s + 7.87e4·27-s − 8.99e4·28-s + ⋯
 L(s)  = 1 + 0.618·2-s + 1.15·3-s − 0.539·4-s + 0.894·5-s + 0.714·6-s + 1.43·7-s − 0.285·8-s + 9-s + 0.553·10-s + 0.781·11-s − 0.622·12-s − 1.13·13-s + 0.889·14-s + 1.03·15-s − 0.296·16-s − 0.271·17-s + 0.618·18-s − 1.65·19-s − 0.482·20-s + 1.65·21-s + 0.483·22-s + 1.57·23-s − 0.329·24-s + 3/5·25-s − 0.702·26-s + 0.769·27-s − 0.774·28-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$225$$    =    $$3^{2} \cdot 5^{2}$$ $$\varepsilon$$ = $1$ motivic weight = $$7$$ character : induced by $\chi_{15} (1, \cdot )$ primitive : no self-dual : yes analytic rank = 0 Selberg data = $(4,\ 225,\ (\ :7/2, 7/2),\ 1)$ $L(4)$ $\approx$ $4.47880$ $L(\frac12)$ $\approx$ $4.47880$ $L(\frac{9}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{3,\;5\}$, $$F_p$$ is a polynomial of degree 4. If $p \in \{3,\;5\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad3$C_1$ $$( 1 - p^{3} T )^{2}$$
5$C_1$ $$( 1 - p^{3} T )^{2}$$
good2$D_{4}$ $$1 - 7 T + 59 p T^{2} - 7 p^{7} T^{3} + p^{14} T^{4}$$
7$D_{4}$ $$1 - 1304 T + 1601006 T^{2} - 1304 p^{7} T^{3} + p^{14} T^{4}$$
11$D_{4}$ $$1 - 3448 T + 9598294 T^{2} - 3448 p^{7} T^{3} + p^{14} T^{4}$$
13$D_{4}$ $$1 + 8988 T + 43676926 T^{2} + 8988 p^{7} T^{3} + p^{14} T^{4}$$
17$D_{4}$ $$1 + 5492 T + 533727862 T^{2} + 5492 p^{7} T^{3} + p^{14} T^{4}$$
19$D_{4}$ $$1 + 49584 T + 1767259558 T^{2} + 49584 p^{7} T^{3} + p^{14} T^{4}$$
23$D_{4}$ $$1 - 91848 T + 4843394254 T^{2} - 91848 p^{7} T^{3} + p^{14} T^{4}$$
29$D_{4}$ $$1 - 6268 p T + 32439203278 T^{2} - 6268 p^{8} T^{3} + p^{14} T^{4}$$
31$D_{4}$ $$1 - 304232 T + 77458297022 T^{2} - 304232 p^{7} T^{3} + p^{14} T^{4}$$
37$D_{4}$ $$1 + 502316 T + 221684315886 T^{2} + 502316 p^{7} T^{3} + p^{14} T^{4}$$
41$D_{4}$ $$1 - 631172 T + 420346017142 T^{2} - 631172 p^{7} T^{3} + p^{14} T^{4}$$
43$D_{4}$ $$1 - 353640 T + 567251429590 T^{2} - 353640 p^{7} T^{3} + p^{14} T^{4}$$
47$D_{4}$ $$1 + 467480 T + 1062629128990 T^{2} + 467480 p^{7} T^{3} + p^{14} T^{4}$$
53$D_{4}$ $$1 + 568052 T + 2403786403294 T^{2} + 568052 p^{7} T^{3} + p^{14} T^{4}$$
59$D_{4}$ $$1 - 287224 T + 1627391637238 T^{2} - 287224 p^{7} T^{3} + p^{14} T^{4}$$
61$D_{4}$ $$1 + 2514180 T + 7865442419758 T^{2} + 2514180 p^{7} T^{3} + p^{14} T^{4}$$
67$D_{4}$ $$1 + 5073832 T + 16021265240102 T^{2} + 5073832 p^{7} T^{3} + p^{14} T^{4}$$
71$D_{4}$ $$1 + 3748816 T + 20824804809646 T^{2} + 3748816 p^{7} T^{3} + p^{14} T^{4}$$
73$D_{4}$ $$1 + 1477212 T - 3158190986 T^{2} + 1477212 p^{7} T^{3} + p^{14} T^{4}$$
79$D_{4}$ $$1 + 4627720 T + 42789559383518 T^{2} + 4627720 p^{7} T^{3} + p^{14} T^{4}$$
83$D_{4}$ $$1 + 6072936 T + 62224060120582 T^{2} + 6072936 p^{7} T^{3} + p^{14} T^{4}$$
89$D_{4}$ $$1 - 16516356 T + 156597055746838 T^{2} - 16516356 p^{7} T^{3} + p^{14} T^{4}$$
97$D_{4}$ $$1 - 2723428 T + 135845063471622 T^{2} - 2723428 p^{7} T^{3} + p^{14} T^{4}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}