Properties

 Label 4-6615-1.1-c1e2-0-3 Degree $4$ Conductor $6615$ Sign $-1$ Analytic cond. $0.421778$ Root an. cond. $0.805881$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $1$

Origins

Dirichlet series

 L(s)  = 1 − 2-s − 3-s − 2·5-s + 6-s − 8-s − 2·9-s + 2·10-s − 11-s + 2·13-s + 2·15-s − 16-s − 4·17-s + 2·18-s − 4·19-s + 22-s − 6·23-s + 24-s + 2·25-s − 2·26-s + 5·27-s − 2·29-s − 2·30-s − 2·31-s + 6·32-s + 33-s + 4·34-s + 37-s + ⋯
 L(s)  = 1 − 0.707·2-s − 0.577·3-s − 0.894·5-s + 0.408·6-s − 0.353·8-s − 2/3·9-s + 0.632·10-s − 0.301·11-s + 0.554·13-s + 0.516·15-s − 1/4·16-s − 0.970·17-s + 0.471·18-s − 0.917·19-s + 0.213·22-s − 1.25·23-s + 0.204·24-s + 2/5·25-s − 0.392·26-s + 0.962·27-s − 0.371·29-s − 0.365·30-s − 0.359·31-s + 1.06·32-s + 0.174·33-s + 0.685·34-s + 0.164·37-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 6615 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 6615 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

 Degree: $$4$$ Conductor: $$6615$$    =    $$3^{3} \cdot 5 \cdot 7^{2}$$ Sign: $-1$ Analytic conductor: $$0.421778$$ Root analytic conductor: $$0.805881$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: $\chi_{6615} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(4,\ 6615,\ (\ :1/2, 1/2),\ -1)$$

Particular Values

 $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ 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$C_2$ $$1 + T + p T^{2}$$
5$C_1$$\times$$C_2$ $$( 1 - T )( 1 + 3 T + p T^{2} )$$
7$C_2$ $$1 + p T^{2}$$
good2$D_{4}$ $$1 + T + T^{2} + p T^{3} + p^{2} T^{4}$$
11$C_2$$\times$$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 3 T + p T^{2} )$$
13$C_2$$\times$$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
17$D_{4}$ $$1 + 4 T + 25 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
19$D_{4}$ $$1 + 4 T + 21 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
23$D_{4}$ $$1 + 6 T + 31 T^{2} + 6 p T^{3} + p^{2} T^{4}$$
29$D_{4}$ $$1 + 2 T - 14 T^{2} + 2 p T^{3} + p^{2} T^{4}$$
31$C_2$$\times$$C_2$ $$( 1 - T + p T^{2} )( 1 + 3 T + p T^{2} )$$
37$D_{4}$ $$1 - T - 24 T^{2} - p T^{3} + p^{2} T^{4}$$
41$C_2$$\times$$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
43$C_2$$\times$$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
47$D_{4}$ $$1 - 5 T + 46 T^{2} - 5 p T^{3} + p^{2} T^{4}$$
53$D_{4}$ $$1 - 2 T + 67 T^{2} - 2 p T^{3} + p^{2} T^{4}$$
59$D_{4}$ $$1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4}$$
61$D_{4}$ $$1 - 2 T - 81 T^{2} - 2 p T^{3} + p^{2} T^{4}$$
67$D_{4}$ $$1 - 11 T + 68 T^{2} - 11 p T^{3} + p^{2} T^{4}$$
71$D_{4}$ $$1 + 2 T - 50 T^{2} + 2 p T^{3} + p^{2} T^{4}$$
73$D_{4}$ $$1 + T + p T^{3} + p^{2} T^{4}$$
79$C_2$$\times$$C_2$ $$( 1 - 9 T + p T^{2} )( 1 + T + p T^{2} )$$
83$D_{4}$ $$1 + 3 T + 82 T^{2} + 3 p T^{3} + p^{2} T^{4}$$
89$C_2$$\times$$C_2$ $$( 1 - 18 T + p T^{2} )( 1 + 15 T + p T^{2} )$$
97$C_2$$\times$$C_2$ $$( 1 - 14 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$