Properties

 Label 4-1575e2-1.1-c3e2-0-1 Degree $4$ Conductor $2480625$ Sign $1$ Analytic cond. $8635.61$ Root an. cond. $9.63991$ Motivic weight $3$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

Origins of factors

Dirichlet series

 L(s)  = 1 − 4·2-s + 4-s + 14·7-s + 24·8-s + 92·11-s − 8·13-s − 56·14-s − 47·16-s − 44·17-s − 108·19-s − 368·22-s − 320·23-s + 32·26-s + 14·28-s + 236·29-s − 60·31-s + 52·32-s + 176·34-s − 204·37-s + 432·38-s − 44·41-s − 136·43-s + 92·44-s + 1.28e3·46-s + 400·47-s + 147·49-s − 8·52-s + ⋯
 L(s)  = 1 − 1.41·2-s + 1/8·4-s + 0.755·7-s + 1.06·8-s + 2.52·11-s − 0.170·13-s − 1.06·14-s − 0.734·16-s − 0.627·17-s − 1.30·19-s − 3.56·22-s − 2.90·23-s + 0.241·26-s + 0.0944·28-s + 1.51·29-s − 0.347·31-s + 0.287·32-s + 0.887·34-s − 0.906·37-s + 1.84·38-s − 0.167·41-s − 0.482·43-s + 0.315·44-s + 4.10·46-s + 1.24·47-s + 3/7·49-s − 0.0213·52-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2480625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2480625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

 Degree: $$4$$ Conductor: $$2480625$$    =    $$3^{4} \cdot 5^{4} \cdot 7^{2}$$ Sign: $1$ Analytic conductor: $$8635.61$$ Root analytic conductor: $$9.63991$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{1575} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 2480625,\ (\ :3/2, 3/2),\ 1)$$

Particular Values

 $$L(2)$$ $$\approx$$ $$1.050428496$$ $$L(\frac12)$$ $$\approx$$ $$1.050428496$$ $$L(\frac{5}{2})$$ 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 $$1$$
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$D_{4}$ $$1 - 2448 T + 3286542 T^{2} - 2448 p^{3} T^{3} + p^{6} T^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$