# Properties

 Degree 4 Conductor $2^{2}$ Sign $1$ Motivic weight 41 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 2.09e6·2-s + 8.86e9·3-s + 3.29e12·4-s + 9.75e13·5-s + 1.85e16·6-s + 2.17e17·7-s + 4.61e18·8-s + 2.37e19·9-s + 2.04e20·10-s + 1.10e20·11-s + 2.92e22·12-s + 1.73e23·13-s + 4.55e23·14-s + 8.65e23·15-s + 6.04e24·16-s − 1.02e25·17-s + 4.98e25·18-s + 2.06e26·19-s + 3.21e26·20-s + 1.92e27·21-s + 2.31e26·22-s + 1.49e28·23-s + 4.08e28·24-s + 8.05e27·25-s + 3.63e29·26-s + 4.87e28·27-s + 7.16e29·28-s + ⋯
 L(s)  = 1 + 1.41·2-s + 1.46·3-s + 3/2·4-s + 0.457·5-s + 2.07·6-s + 1.02·7-s + 1.41·8-s + 0.652·9-s + 0.647·10-s + 0.0495·11-s + 2.20·12-s + 2.52·13-s + 1.45·14-s + 0.671·15-s + 5/4·16-s − 0.613·17-s + 0.922·18-s + 1.25·19-s + 0.686·20-s + 1.50·21-s + 0.0700·22-s + 1.81·23-s + 2.07·24-s + 0.177·25-s + 3.57·26-s + 0.221·27-s + 1.54·28-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(42-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+41/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$4$$    =    $$2^{2}$$ $$\varepsilon$$ = $1$ motivic weight = $$41$$ character : induced by $\chi_{2} (1, \cdot )$ primitive : no self-dual : yes analytic rank = 0 Selberg data = $(4,\ 4,\ (\ :41/2, 41/2),\ 1)$ $L(21)$ $\approx$ $18.1697$ $L(\frac12)$ $\approx$ $18.1697$ $L(\frac{43}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \neq 2$, $$F_p$$ is a polynomial of degree 4. If $p = 2$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_1$ $$( 1 - p^{20} T )^{2}$$
good3$D_{4}$ $$1 - 984816392 p^{2} T + 927466955744998 p^{10} T^{2} - 984816392 p^{43} T^{3} + p^{82} T^{4}$$
5$D_{4}$ $$1 - 19519836865116 p T +$$$$47\!\cdots\!46$$$$p^{5} T^{2} - 19519836865116 p^{42} T^{3} + p^{82} T^{4}$$
7$D_{4}$ $$1 - 4430751068838544 p^{2} T +$$$$79\!\cdots\!86$$$$p^{7} T^{2} - 4430751068838544 p^{43} T^{3} + p^{82} T^{4}$$
11$D_{4}$ $$1 - 10054043771358142344 p T +$$$$64\!\cdots\!06$$$$p^{3} T^{2} - 10054043771358142344 p^{42} T^{3} + p^{82} T^{4}$$
13$D_{4}$ $$1 -$$$$13\!\cdots\!36$$$$p T +$$$$76\!\cdots\!06$$$$p^{3} T^{2} -$$$$13\!\cdots\!36$$$$p^{42} T^{3} + p^{82} T^{4}$$
17$D_{4}$ $$1 +$$$$10\!\cdots\!04$$$$T +$$$$34\!\cdots\!14$$$$p T^{2} +$$$$10\!\cdots\!04$$$$p^{41} T^{3} + p^{82} T^{4}$$
19$D_{4}$ $$1 -$$$$10\!\cdots\!00$$$$p T +$$$$65\!\cdots\!82$$$$p^{3} T^{2} -$$$$10\!\cdots\!00$$$$p^{42} T^{3} + p^{82} T^{4}$$
23$D_{4}$ $$1 -$$$$14\!\cdots\!68$$$$T +$$$$82\!\cdots\!74$$$$p T^{2} -$$$$14\!\cdots\!68$$$$p^{41} T^{3} + p^{82} T^{4}$$
29$D_{4}$ $$1 +$$$$47\!\cdots\!80$$$$p T +$$$$47\!\cdots\!02$$$$p T^{2} +$$$$47\!\cdots\!80$$$$p^{42} T^{3} + p^{82} T^{4}$$
31$D_{4}$ $$1 +$$$$11\!\cdots\!36$$$$p T +$$$$10\!\cdots\!66$$$$p^{2} T^{2} +$$$$11\!\cdots\!36$$$$p^{42} T^{3} + p^{82} T^{4}$$
37$D_{4}$ $$1 -$$$$32\!\cdots\!28$$$$p T +$$$$31\!\cdots\!42$$$$p^{2} T^{2} -$$$$32\!\cdots\!28$$$$p^{42} T^{3} + p^{82} T^{4}$$
41$D_{4}$ $$1 -$$$$12\!\cdots\!64$$$$T +$$$$75\!\cdots\!06$$$$T^{2} -$$$$12\!\cdots\!64$$$$p^{41} T^{3} + p^{82} T^{4}$$
43$D_{4}$ $$1 +$$$$60\!\cdots\!52$$$$T +$$$$20\!\cdots\!62$$$$T^{2} +$$$$60\!\cdots\!52$$$$p^{41} T^{3} + p^{82} T^{4}$$
47$D_{4}$ $$1 +$$$$25\!\cdots\!64$$$$T +$$$$84\!\cdots\!18$$$$T^{2} +$$$$25\!\cdots\!64$$$$p^{41} T^{3} + p^{82} T^{4}$$
53$D_{4}$ $$1 +$$$$30\!\cdots\!52$$$$T +$$$$11\!\cdots\!82$$$$T^{2} +$$$$30\!\cdots\!52$$$$p^{41} T^{3} + p^{82} T^{4}$$
59$D_{4}$ $$1 +$$$$32\!\cdots\!40$$$$T +$$$$78\!\cdots\!18$$$$T^{2} +$$$$32\!\cdots\!40$$$$p^{41} T^{3} + p^{82} T^{4}$$
61$D_{4}$ $$1 +$$$$44\!\cdots\!36$$$$T +$$$$17\!\cdots\!46$$$$T^{2} +$$$$44\!\cdots\!36$$$$p^{41} T^{3} + p^{82} T^{4}$$
67$D_{4}$ $$1 -$$$$55\!\cdots\!56$$$$T +$$$$15\!\cdots\!18$$$$T^{2} -$$$$55\!\cdots\!56$$$$p^{41} T^{3} + p^{82} T^{4}$$
71$D_{4}$ $$1 +$$$$48\!\cdots\!16$$$$T +$$$$10\!\cdots\!06$$$$T^{2} +$$$$48\!\cdots\!16$$$$p^{41} T^{3} + p^{82} T^{4}$$
73$D_{4}$ $$1 +$$$$48\!\cdots\!12$$$$T +$$$$10\!\cdots\!82$$$$T^{2} +$$$$48\!\cdots\!12$$$$p^{41} T^{3} + p^{82} T^{4}$$
79$D_{4}$ $$1 +$$$$90\!\cdots\!80$$$$T +$$$$14\!\cdots\!58$$$$T^{2} +$$$$90\!\cdots\!80$$$$p^{41} T^{3} + p^{82} T^{4}$$
83$D_{4}$ $$1 -$$$$12\!\cdots\!68$$$$T +$$$$95\!\cdots\!22$$$$T^{2} -$$$$12\!\cdots\!68$$$$p^{41} T^{3} + p^{82} T^{4}$$
89$D_{4}$ $$1 -$$$$26\!\cdots\!80$$$$T +$$$$16\!\cdots\!78$$$$T^{2} -$$$$26\!\cdots\!80$$$$p^{41} T^{3} + p^{82} T^{4}$$
97$D_{4}$ $$1 +$$$$31\!\cdots\!04$$$$T +$$$$59\!\cdots\!98$$$$T^{2} +$$$$31\!\cdots\!04$$$$p^{41} T^{3} + p^{82} T^{4}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}