# Properties

 Label 4-1-1.1-c31e2-0-0 Degree $4$ Conductor $1$ Sign $1$ Analytic cond. $37.0602$ Root an. cond. $2.46732$ Motivic weight $31$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 3.99e4·2-s + 1.73e7·3-s − 4.62e8·4-s − 1.93e10·5-s + 6.93e11·6-s + 3.02e13·7-s − 1.49e13·8-s − 5.17e14·9-s − 7.74e14·10-s − 7.78e15·11-s − 8.03e15·12-s + 7.47e16·13-s + 1.20e18·14-s − 3.36e17·15-s − 8.28e17·16-s + 1.72e19·17-s − 2.06e19·18-s − 1.23e19·19-s + 8.97e18·20-s + 5.25e20·21-s − 3.10e20·22-s + 1.89e21·23-s − 2.60e20·24-s − 3.72e21·25-s + 2.98e21·26-s − 1.24e22·27-s − 1.40e22·28-s + ⋯
 L(s)  = 1 + 0.862·2-s + 0.698·3-s − 0.215·4-s − 0.284·5-s + 0.602·6-s + 2.40·7-s − 0.150·8-s − 0.837·9-s − 0.245·10-s − 0.561·11-s − 0.150·12-s + 0.404·13-s + 2.07·14-s − 0.198·15-s − 0.179·16-s + 1.45·17-s − 0.722·18-s − 0.186·19-s + 0.0612·20-s + 1.68·21-s − 0.484·22-s + 1.48·23-s − 0.105·24-s − 0.800·25-s + 0.349·26-s − 0.813·27-s − 0.519·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s)^{2} \, L(s)\cr=\mathstrut & \,\Lambda(32-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+31/2)^{2} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$1$$ Sign: $1$ Analytic conductor: $$37.0602$$ Root analytic conductor: $$2.46732$$ Motivic weight: $$31$$ Rational: yes Arithmetic: yes Character: $\chi_{1} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 1,\ (\ :31/2, 31/2),\ 1)$$

## Particular Values

 $$L(16)$$ $$\approx$$ $$4.421604976$$ $$L(\frac12)$$ $$\approx$$ $$4.421604976$$ $$L(\frac{33}{2})$$ 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)$
good2$D_{4}$ $$1 - 4995 p^{3} T + 2011345 p^{10} T^{2} - 4995 p^{34} T^{3} + p^{62} T^{4}$$
3$D_{4}$ $$1 - 214360 p^{4} T + 13870620790 p^{10} T^{2} - 214360 p^{35} T^{3} + p^{62} T^{4}$$
5$D_{4}$ $$1 + 3878243604 p T + 1313821413051106982 p^{5} T^{2} + 3878243604 p^{32} T^{3} + p^{62} T^{4}$$
7$D_{4}$ $$1 - 617500562800 p^{2} T +$$$$31\!\cdots\!50$$$$p^{5} T^{2} - 617500562800 p^{33} T^{3} + p^{62} T^{4}$$
11$D_{4}$ $$1 + 7782353745118776 T +$$$$32\!\cdots\!46$$$$p^{2} T^{2} + 7782353745118776 p^{31} T^{3} + p^{62} T^{4}$$
13$D_{4}$ $$1 - 5746842542327740 p T +$$$$55\!\cdots\!10$$$$p^{3} T^{2} - 5746842542327740 p^{32} T^{3} + p^{62} T^{4}$$
17$D_{4}$ $$1 - 17224607828987089380 T +$$$$20\!\cdots\!70$$$$p T^{2} - 17224607828987089380 p^{31} T^{3} + p^{62} T^{4}$$
19$D_{4}$ $$1 + 651082280422219160 p T +$$$$24\!\cdots\!58$$$$p^{2} T^{2} + 651082280422219160 p^{32} T^{3} + p^{62} T^{4}$$
23$D_{4}$ $$1 - 82493253999561357360 p T +$$$$61\!\cdots\!70$$$$p^{2} T^{2} - 82493253999561357360 p^{32} T^{3} + p^{62} T^{4}$$
29$D_{4}$ $$1 -$$$$44\!\cdots\!60$$$$p T +$$$$98\!\cdots\!38$$$$p^{2} T^{2} -$$$$44\!\cdots\!60$$$$p^{32} T^{3} + p^{62} T^{4}$$
31$D_{4}$ $$1 -$$$$12\!\cdots\!64$$$$T +$$$$22\!\cdots\!86$$$$T^{2} -$$$$12\!\cdots\!64$$$$p^{31} T^{3} + p^{62} T^{4}$$
37$D_{4}$ $$1 +$$$$83\!\cdots\!60$$$$T +$$$$56\!\cdots\!70$$$$T^{2} +$$$$83\!\cdots\!60$$$$p^{31} T^{3} + p^{62} T^{4}$$
41$D_{4}$ $$1 -$$$$87\!\cdots\!84$$$$T +$$$$12\!\cdots\!46$$$$T^{2} -$$$$87\!\cdots\!84$$$$p^{31} T^{3} + p^{62} T^{4}$$
43$D_{4}$ $$1 +$$$$18\!\cdots\!00$$$$T +$$$$64\!\cdots\!50$$$$T^{2} +$$$$18\!\cdots\!00$$$$p^{31} T^{3} + p^{62} T^{4}$$
47$D_{4}$ $$1 -$$$$95\!\cdots\!20$$$$T +$$$$15\!\cdots\!10$$$$T^{2} -$$$$95\!\cdots\!20$$$$p^{31} T^{3} + p^{62} T^{4}$$
53$D_{4}$ $$1 -$$$$19\!\cdots\!60$$$$T +$$$$57\!\cdots\!10$$$$T^{2} -$$$$19\!\cdots\!60$$$$p^{31} T^{3} + p^{62} T^{4}$$
59$D_{4}$ $$1 +$$$$19\!\cdots\!20$$$$T +$$$$10\!\cdots\!18$$$$T^{2} +$$$$19\!\cdots\!20$$$$p^{31} T^{3} + p^{62} T^{4}$$
61$D_{4}$ $$1 +$$$$12\!\cdots\!76$$$$T +$$$$80\!\cdots\!66$$$$T^{2} +$$$$12\!\cdots\!76$$$$p^{31} T^{3} + p^{62} T^{4}$$
67$D_{4}$ $$1 +$$$$96\!\cdots\!20$$$$T +$$$$81\!\cdots\!90$$$$T^{2} +$$$$96\!\cdots\!20$$$$p^{31} T^{3} + p^{62} T^{4}$$
71$D_{4}$ $$1 -$$$$55\!\cdots\!44$$$$T +$$$$32\!\cdots\!26$$$$T^{2} -$$$$55\!\cdots\!44$$$$p^{31} T^{3} + p^{62} T^{4}$$
73$D_{4}$ $$1 -$$$$62\!\cdots\!80$$$$T +$$$$12\!\cdots\!30$$$$T^{2} -$$$$62\!\cdots\!80$$$$p^{31} T^{3} + p^{62} T^{4}$$
79$D_{4}$ $$1 +$$$$11\!\cdots\!60$$$$T +$$$$80\!\cdots\!58$$$$T^{2} +$$$$11\!\cdots\!60$$$$p^{31} T^{3} + p^{62} T^{4}$$
83$D_{4}$ $$1 +$$$$26\!\cdots\!60$$$$T +$$$$53\!\cdots\!90$$$$T^{2} +$$$$26\!\cdots\!60$$$$p^{31} T^{3} + p^{62} T^{4}$$
89$D_{4}$ $$1 +$$$$21\!\cdots\!80$$$$T +$$$$47\!\cdots\!78$$$$T^{2} +$$$$21\!\cdots\!80$$$$p^{31} T^{3} + p^{62} T^{4}$$
97$D_{4}$ $$1 +$$$$90\!\cdots\!80$$$$T +$$$$97\!\cdots\!10$$$$T^{2} +$$$$90\!\cdots\!80$$$$p^{31} T^{3} + p^{62} T^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$