Properties

 Label 4-2205e2-1.1-c3e2-0-9 Degree $4$ Conductor $4862025$ Sign $1$ Analytic cond. $16925.8$ Root an. cond. $11.4061$ Motivic weight $3$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $2$

Origins of factors

Dirichlet series

 L(s)  = 1 − 2·2-s − 2·4-s + 10·5-s − 20·10-s − 66·11-s − 10·13-s − 20·16-s + 70·17-s − 140·19-s − 20·20-s + 132·22-s + 16·23-s + 75·25-s + 20·26-s + 258·29-s + 20·31-s + 200·32-s − 140·34-s + 328·37-s + 280·38-s + 300·41-s − 116·43-s + 132·44-s − 32·46-s − 30·47-s − 150·50-s + 20·52-s + ⋯
 L(s)  = 1 − 0.707·2-s − 1/4·4-s + 0.894·5-s − 0.632·10-s − 1.80·11-s − 0.213·13-s − 0.312·16-s + 0.998·17-s − 1.69·19-s − 0.223·20-s + 1.27·22-s + 0.145·23-s + 3/5·25-s + 0.150·26-s + 1.65·29-s + 0.115·31-s + 1.10·32-s − 0.706·34-s + 1.45·37-s + 1.19·38-s + 1.14·41-s − 0.411·43-s + 0.452·44-s − 0.102·46-s − 0.0931·47-s − 0.424·50-s + 0.0533·52-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 4862025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 4862025 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

 Degree: $$4$$ Conductor: $$4862025$$    =    $$3^{4} \cdot 5^{2} \cdot 7^{4}$$ Sign: $1$ Analytic conductor: $$16925.8$$ Root analytic conductor: $$11.4061$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$2$$ Selberg data: $$(4,\ 4862025,\ (\ :3/2, 3/2),\ 1)$$

Particular Values

 $$L(2)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$\Gal(F_p)$$F_p(T)$
bad3 $$1$$
5$C_1$ $$( 1 - p T )^{2}$$
7 $$1$$
good2$D_{4}$ $$1 + p T + 3 p T^{2} + p^{4} T^{3} + p^{6} T^{4}$$
11$D_{4}$ $$1 + 6 p T + 325 p T^{2} + 6 p^{4} T^{3} + p^{6} T^{4}$$
13$D_{4}$ $$1 + 10 T + 19 T^{2} + 10 p^{3} T^{3} + p^{6} T^{4}$$
17$D_{4}$ $$1 - 70 T + 6651 T^{2} - 70 p^{3} T^{3} + p^{6} T^{4}$$
19$D_{4}$ $$1 + 140 T + 14218 T^{2} + 140 p^{3} T^{3} + p^{6} T^{4}$$
23$D_{4}$ $$1 - 16 T + 15774 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4}$$
29$D_{4}$ $$1 - 258 T + 40075 T^{2} - 258 p^{3} T^{3} + p^{6} T^{4}$$
31$D_{4}$ $$1 - 20 T + 20082 T^{2} - 20 p^{3} T^{3} + p^{6} T^{4}$$
37$D_{4}$ $$1 - 328 T + 106906 T^{2} - 328 p^{3} T^{3} + p^{6} T^{4}$$
41$D_{4}$ $$1 - 300 T + 50342 T^{2} - 300 p^{3} T^{3} + p^{6} T^{4}$$
43$D_{4}$ $$1 + 116 T + 160794 T^{2} + 116 p^{3} T^{3} + p^{6} T^{4}$$
47$D_{4}$ $$1 + 30 T + 190271 T^{2} + 30 p^{3} T^{3} + p^{6} T^{4}$$
53$D_{4}$ $$1 + 540 T + 212254 T^{2} + 540 p^{3} T^{3} + p^{6} T^{4}$$
59$D_{4}$ $$1 - 380 T + 429258 T^{2} - 380 p^{3} T^{3} + p^{6} T^{4}$$
61$D_{4}$ $$1 + 1080 T + 705962 T^{2} + 1080 p^{3} T^{3} + p^{6} T^{4}$$
67$D_{4}$ $$1 - 468 T + 554906 T^{2} - 468 p^{3} T^{3} + p^{6} T^{4}$$
71$D_{4}$ $$1 - 1056 T + 949550 T^{2} - 1056 p^{3} T^{3} + p^{6} T^{4}$$
73$D_{4}$ $$1 - 860 T + 522934 T^{2} - 860 p^{3} T^{3} + p^{6} T^{4}$$
79$D_{4}$ $$1 - 2 p T - 339825 T^{2} - 2 p^{4} T^{3} + p^{6} T^{4}$$
83$D_{4}$ $$1 - 40 T + 703974 T^{2} - 40 p^{3} T^{3} + p^{6} T^{4}$$
89$D_{4}$ $$1 + 240 T - 164062 T^{2} + 240 p^{3} T^{3} + p^{6} T^{4}$$
97$D_{4}$ $$1 + 1630 T + 2133171 T^{2} + 1630 p^{3} T^{3} + p^{6} T^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$