Properties

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

Origins of factors

Dirichlet series

 L(s)  = 1 + 3·2-s + 6·3-s + 4-s − 10·5-s + 18·6-s + 14·7-s + 3·8-s + 27·9-s − 30·10-s + 62·11-s + 6·12-s − 6·13-s + 42·14-s − 60·15-s + 9·16-s + 40·17-s + 81·18-s − 122·19-s − 10·20-s + 84·21-s + 186·22-s + 16·23-s + 18·24-s + 75·25-s − 18·26-s + 108·27-s + 14·28-s + ⋯
 L(s)  = 1 + 1.06·2-s + 1.15·3-s + 1/8·4-s − 0.894·5-s + 1.22·6-s + 0.755·7-s + 0.132·8-s + 9-s − 0.948·10-s + 1.69·11-s + 0.144·12-s − 0.128·13-s + 0.801·14-s − 1.03·15-s + 9/64·16-s + 0.570·17-s + 1.06·18-s − 1.47·19-s − 0.111·20-s + 0.872·21-s + 1.80·22-s + 0.145·23-s + 0.153·24-s + 3/5·25-s − 0.135·26-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 = $$0$$ Selberg data = $$(4,\ 11025,\ (\ :3/2, 3/2),\ 1)$$ $$L(2)$$ $$\approx$$ $$5.27205$$ $$L(\frac12)$$ $$\approx$$ $$5.27205$$ $$L(\frac{5}{2})$$ 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$D_{4}$ $$1 - 3 T + p^{3} T^{2} - 3 p^{3} T^{3} + p^{6} T^{4}$$
11$D_{4}$ $$1 - 62 T + 3582 T^{2} - 62 p^{3} T^{3} + p^{6} T^{4}$$
13$D_{4}$ $$1 + 6 T + 3378 T^{2} + 6 p^{3} T^{3} + p^{6} T^{4}$$
17$D_{4}$ $$1 - 40 T + 8750 T^{2} - 40 p^{3} T^{3} + p^{6} T^{4}$$
19$D_{4}$ $$1 + 122 T + 17398 T^{2} + 122 p^{3} T^{3} + p^{6} T^{4}$$
23$D_{4}$ $$1 - 16 T + 34 p T^{2} - 16 p^{3} T^{3} + p^{6} T^{4}$$
29$D_{4}$ $$1 - 352 T + 78278 T^{2} - 352 p^{3} T^{3} + p^{6} T^{4}$$
31$D_{4}$ $$1 - 66 T + 45870 T^{2} - 66 p^{3} T^{3} + p^{6} T^{4}$$
37$D_{4}$ $$1 + 188 T + 44542 T^{2} + 188 p^{3} T^{3} + p^{6} T^{4}$$
41$D_{4}$ $$1 - 16 T + 18350 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4}$$
43$D_{4}$ $$1 + 396 T + 2226 p T^{2} + 396 p^{3} T^{3} + p^{6} T^{4}$$
47$D_{4}$ $$1 + 4 p T + 15582 T^{2} + 4 p^{4} T^{3} + p^{6} T^{4}$$
53$D_{4}$ $$1 - 982 T + 504354 T^{2} - 982 p^{3} T^{3} + p^{6} T^{4}$$
59$D_{4}$ $$1 - 516 T + 418118 T^{2} - 516 p^{3} T^{3} + p^{6} T^{4}$$
61$D_{4}$ $$1 + 880 T + 575238 T^{2} + 880 p^{3} T^{3} + p^{6} T^{4}$$
67$D_{4}$ $$1 + 356 T + 100374 T^{2} + 356 p^{3} T^{3} + p^{6} T^{4}$$
71$D_{4}$ $$1 - 310 T + 664038 T^{2} - 310 p^{3} T^{3} + p^{6} T^{4}$$
73$D_{4}$ $$1 - 326 T + 789802 T^{2} - 326 p^{3} T^{3} + p^{6} T^{4}$$
79$D_{4}$ $$1 - 1832 T + 1824478 T^{2} - 1832 p^{3} T^{3} + p^{6} T^{4}$$
83$D_{4}$ $$1 + 680 T + 744870 T^{2} + 680 p^{3} T^{3} + p^{6} T^{4}$$
89$D_{4}$ $$1 - 796 T + 411158 T^{2} - 796 p^{3} T^{3} + p^{6} T^{4}$$
97$D_{4}$ $$1 + 670 T + 1145410 T^{2} + 670 p^{3} T^{3} + p^{6} T^{4}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}