# Properties

 Label 4-2496e2-1.1-c3e2-0-13 Degree $4$ Conductor $6230016$ Sign $1$ Analytic cond. $21688.0$ Root an. cond. $12.1354$ Motivic weight $3$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $2$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 6·3-s + 4·5-s − 20·7-s + 27·9-s + 60·11-s + 26·13-s − 24·15-s − 84·17-s + 60·19-s + 120·21-s − 72·23-s − 210·25-s − 108·27-s + 124·29-s − 108·31-s − 360·33-s − 80·35-s − 36·37-s − 156·39-s − 52·41-s + 32·43-s + 108·45-s − 428·47-s − 134·49-s + 504·51-s − 380·53-s + 240·55-s + ⋯
 L(s)  = 1 − 1.15·3-s + 0.357·5-s − 1.07·7-s + 9-s + 1.64·11-s + 0.554·13-s − 0.413·15-s − 1.19·17-s + 0.724·19-s + 1.24·21-s − 0.652·23-s − 1.67·25-s − 0.769·27-s + 0.794·29-s − 0.625·31-s − 1.89·33-s − 0.386·35-s − 0.159·37-s − 0.640·39-s − 0.198·41-s + 0.113·43-s + 0.357·45-s − 1.32·47-s − 0.390·49-s + 1.38·51-s − 0.984·53-s + 0.588·55-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 6230016 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 6230016 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$6230016$$    =    $$2^{12} \cdot 3^{2} \cdot 13^{2}$$ Sign: $1$ Analytic conductor: $$21688.0$$ Root analytic conductor: $$12.1354$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$2$$ Selberg data: $$(4,\ 6230016,\ (\ :3/2, 3/2),\ 1)$$

## 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)$
bad2 $$1$$
3$C_1$ $$( 1 + p T )^{2}$$
13$C_1$ $$( 1 - p T )^{2}$$
good5$D_{4}$ $$1 - 4 T + 226 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4}$$
7$D_{4}$ $$1 + 20 T + 534 T^{2} + 20 p^{3} T^{3} + p^{6} T^{4}$$
11$D_{4}$ $$1 - 60 T + 3114 T^{2} - 60 p^{3} T^{3} + p^{6} T^{4}$$
17$D_{4}$ $$1 + 84 T + 10582 T^{2} + 84 p^{3} T^{3} + p^{6} T^{4}$$
19$D_{4}$ $$1 - 60 T + 6526 T^{2} - 60 p^{3} T^{3} + p^{6} T^{4}$$
23$D_{4}$ $$1 + 72 T + 14430 T^{2} + 72 p^{3} T^{3} + p^{6} T^{4}$$
29$D_{4}$ $$1 - 124 T - 1586 T^{2} - 124 p^{3} T^{3} + p^{6} T^{4}$$
31$D_{4}$ $$1 + 108 T - 16154 T^{2} + 108 p^{3} T^{3} + p^{6} T^{4}$$
37$D_{4}$ $$1 + 36 T + 42382 T^{2} + 36 p^{3} T^{3} + p^{6} T^{4}$$
41$D_{4}$ $$1 + 52 T + 76666 T^{2} + 52 p^{3} T^{3} + p^{6} T^{4}$$
43$D_{4}$ $$1 - 32 T + 150198 T^{2} - 32 p^{3} T^{3} + p^{6} T^{4}$$
47$D_{4}$ $$1 + 428 T + 116242 T^{2} + 428 p^{3} T^{3} + p^{6} T^{4}$$
53$D_{4}$ $$1 + 380 T + 258142 T^{2} + 380 p^{3} T^{3} + p^{6} T^{4}$$
59$D_{4}$ $$1 - 1420 T + 882490 T^{2} - 1420 p^{3} T^{3} + p^{6} T^{4}$$
61$D_{4}$ $$1 + 1012 T + 708206 T^{2} + 1012 p^{3} T^{3} + p^{6} T^{4}$$
67$D_{4}$ $$1 - 844 T + 778238 T^{2} - 844 p^{3} T^{3} + p^{6} T^{4}$$
71$D_{4}$ $$1 + 868 T + 888050 T^{2} + 868 p^{3} T^{3} + p^{6} T^{4}$$
73$D_{4}$ $$1 + 60 T + 114886 T^{2} + 60 p^{3} T^{3} + p^{6} T^{4}$$
79$D_{4}$ $$1 - 272 T + 993374 T^{2} - 272 p^{3} T^{3} + p^{6} T^{4}$$
83$D_{4}$ $$1 - 1252 T + 1118698 T^{2} - 1252 p^{3} T^{3} + p^{6} T^{4}$$
89$D_{4}$ $$1 - 572 T + 853306 T^{2} - 572 p^{3} T^{3} + p^{6} T^{4}$$
97$D_{4}$ $$1 - 708 T + 1762390 T^{2} - 708 p^{3} T^{3} + p^{6} T^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$