Properties

 Label 4-501120-1.1-c1e2-0-37 Degree $4$ Conductor $501120$ Sign $-1$ Analytic cond. $31.9518$ Root an. cond. $2.37751$ 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 + 4-s − 5-s + 6-s + 8-s + 9-s − 10-s + 12-s − 15-s + 16-s + 18-s + 2·19-s − 20-s − 12·23-s + 24-s − 2·25-s + 27-s − 9·29-s − 30-s + 32-s + 36-s + 2·38-s − 40-s − 14·43-s − 45-s − 12·46-s + ⋯
 L(s)  = 1 + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.288·12-s − 0.258·15-s + 1/4·16-s + 0.235·18-s + 0.458·19-s − 0.223·20-s − 2.50·23-s + 0.204·24-s − 2/5·25-s + 0.192·27-s − 1.67·29-s − 0.182·30-s + 0.176·32-s + 1/6·36-s + 0.324·38-s − 0.158·40-s − 2.13·43-s − 0.149·45-s − 1.76·46-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 501120 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 501120 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

 Degree: $$4$$ Conductor: $$501120$$    =    $$2^{7} \cdot 3^{3} \cdot 5 \cdot 29$$ Sign: $-1$ Analytic conductor: $$31.9518$$ Root analytic conductor: $$2.37751$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(4,\ 501120,\ (\ :1/2, 1/2),\ -1)$$

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)$
bad2$C_1$ $$1 - T$$
3$C_1$ $$1 - T$$
5$C_1$$\times$$C_2$ $$( 1 - T )( 1 + 2 T + p T^{2} )$$
29$C_1$$\times$$C_2$ $$( 1 + T )( 1 + 8 T + p T^{2} )$$
good7$C_2^2$ $$1 + 2 T^{2} + p^{2} T^{4}$$
11$C_2^2$ $$1 + 10 T^{2} + p^{2} T^{4}$$
13$C_2^2$ $$1 + 2 T^{2} + p^{2} T^{4}$$
17$C_2^2$ $$1 - 6 T^{2} + p^{2} T^{4}$$
19$C_2$$\times$$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + p T^{2} )$$
23$C_2$$\times$$C_2$ $$( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
31$C_2^2$ $$1 - 16 T^{2} + p^{2} T^{4}$$
37$C_2^2$ $$1 + 24 T^{2} + p^{2} T^{4}$$
41$C_2^2$ $$1 + 50 T^{2} + p^{2} T^{4}$$
43$C_2$$\times$$C_2$ $$( 1 + 6 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
47$C_2$$\times$$C_2$ $$( 1 + p T^{2} )( 1 + 6 T + p T^{2} )$$
53$C_2$$\times$$C_2$ $$( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
59$C_2^2$ $$1 + 58 T^{2} + p^{2} T^{4}$$
61$C_2^2$ $$1 + 40 T^{2} + p^{2} T^{4}$$
67$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
71$C_2$$\times$$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + p T^{2} )$$
73$C_2$$\times$$C_2$ $$( 1 + p T^{2} )( 1 + 10 T + p T^{2} )$$
79$C_2^2$ $$1 + 140 T^{2} + p^{2} T^{4}$$
83$C_2^2$ $$1 + 114 T^{2} + p^{2} T^{4}$$
89$C_2^2$ $$1 - 50 T^{2} + p^{2} T^{4}$$
97$C_2$$\times$$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + p T^{2} )$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$