Properties

 Label 4-1386e2-1.1-c1e2-0-41 Degree $4$ Conductor $1920996$ Sign $1$ Analytic cond. $122.484$ Root an. cond. $3.32675$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $2$

Origins of factors

Dirichlet series

 L(s)  = 1 + 2-s + 2·5-s − 5·7-s − 8-s + 2·10-s − 11-s − 14·13-s − 5·14-s − 16-s + 2·17-s − 22-s − 8·23-s + 5·25-s − 14·26-s + 10·29-s − 4·31-s + 2·34-s − 10·35-s − 4·37-s − 2·40-s − 8·41-s − 16·43-s − 8·46-s + 2·47-s + 18·49-s + 5·50-s − 6·53-s + ⋯
 L(s)  = 1 + 0.707·2-s + 0.894·5-s − 1.88·7-s − 0.353·8-s + 0.632·10-s − 0.301·11-s − 3.88·13-s − 1.33·14-s − 1/4·16-s + 0.485·17-s − 0.213·22-s − 1.66·23-s + 25-s − 2.74·26-s + 1.85·29-s − 0.718·31-s + 0.342·34-s − 1.69·35-s − 0.657·37-s − 0.316·40-s − 1.24·41-s − 2.43·43-s − 1.17·46-s + 0.291·47-s + 18/7·49-s + 0.707·50-s − 0.824·53-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1920996 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1920996 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

 Degree: $$4$$ Conductor: $$1920996$$    =    $$2^{2} \cdot 3^{4} \cdot 7^{2} \cdot 11^{2}$$ Sign: $1$ Analytic conductor: $$122.484$$ Root analytic conductor: $$3.32675$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{1386} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$2$$ Selberg data: $$(4,\ 1920996,\ (\ :1/2, 1/2),\ 1)$$

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$C_2$ $$1 - T + T^{2}$$
3 $$1$$
7$C_2$ $$1 + 5 T + p T^{2}$$
11$C_2$ $$1 + T + T^{2}$$
good5$C_2^2$ $$1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4}$$
13$C_2$ $$( 1 + 7 T + p T^{2} )^{2}$$
17$C_2^2$ $$1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4}$$
19$C_2^2$ $$1 - p T^{2} + p^{2} T^{4}$$
23$C_2^2$ $$1 + 8 T + 41 T^{2} + 8 p T^{3} + p^{2} T^{4}$$
29$C_2$ $$( 1 - 5 T + p T^{2} )^{2}$$
31$C_2$ $$( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} )$$
37$C_2^2$ $$1 + 4 T - 21 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
41$C_2$ $$( 1 + 4 T + p T^{2} )^{2}$$
43$C_2$ $$( 1 + 8 T + p T^{2} )^{2}$$
47$C_2^2$ $$1 - 2 T - 43 T^{2} - 2 p T^{3} + p^{2} T^{4}$$
53$C_2^2$ $$1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4}$$
59$C_2^2$ $$1 - 3 T - 50 T^{2} - 3 p T^{3} + p^{2} T^{4}$$
61$C_2$ $$( 1 - 13 T + p T^{2} )( 1 + 14 T + p T^{2} )$$
67$C_2^2$ $$1 + 9 T + 14 T^{2} + 9 p T^{3} + p^{2} T^{4}$$
71$C_2$ $$( 1 - 2 T + p T^{2} )^{2}$$
73$C_2^2$ $$1 + 4 T - 57 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
79$C_2^2$ $$1 + 9 T + 2 T^{2} + 9 p T^{3} + p^{2} T^{4}$$
83$C_2$ $$( 1 + 6 T + p T^{2} )^{2}$$
89$C_2^2$ $$1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4}$$
97$C_2$ $$( 1 - 7 T + p T^{2} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$