# Properties

 Label 4-22e2-1.1-c13e2-0-0 Degree $4$ Conductor $484$ Sign $1$ Analytic cond. $556.526$ Root an. cond. $4.85703$ Motivic weight $13$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 128·2-s + 1.62e3·3-s + 1.22e4·4-s + 7.66e3·5-s − 2.08e5·6-s + 6.37e5·7-s − 1.04e6·8-s + 7.90e5·9-s − 9.81e5·10-s + 3.54e6·11-s + 1.99e7·12-s − 3.78e6·13-s − 8.15e7·14-s + 1.24e7·15-s + 8.38e7·16-s + 3.71e7·17-s − 1.01e8·18-s − 1.02e8·19-s + 9.41e7·20-s + 1.03e9·21-s − 4.53e8·22-s + 1.35e9·23-s − 1.70e9·24-s − 1.11e9·25-s + 4.84e8·26-s + 8.62e8·27-s + 7.82e9·28-s + ⋯
 L(s)  = 1 − 1.41·2-s + 1.28·3-s + 3/2·4-s + 0.219·5-s − 1.82·6-s + 2.04·7-s − 1.41·8-s + 0.495·9-s − 0.310·10-s + 0.603·11-s + 1.93·12-s − 0.217·13-s − 2.89·14-s + 0.282·15-s + 5/4·16-s + 0.373·17-s − 0.700·18-s − 0.499·19-s + 0.329·20-s + 2.63·21-s − 0.852·22-s + 1.90·23-s − 1.82·24-s − 0.914·25-s + 0.307·26-s + 0.428·27-s + 3.06·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 484 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 484 ^{s/2} \, \Gamma_{\C}(s+13/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$484$$    =    $$2^{2} \cdot 11^{2}$$ Sign: $1$ Analytic conductor: $$556.526$$ Root analytic conductor: $$4.85703$$ Motivic weight: $$13$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 484,\ (\ :13/2, 13/2),\ 1)$$

## Particular Values

 $$L(7)$$ $$\approx$$ $$4.116220184$$ $$L(\frac12)$$ $$\approx$$ $$4.116220184$$ $$L(\frac{15}{2})$$ 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$C_1$ $$( 1 + p^{6} T )^{2}$$
11$C_1$ $$( 1 - p^{6} T )^{2}$$
good3$D_{4}$ $$1 - 542 p T + 68657 p^{3} T^{2} - 542 p^{14} T^{3} + p^{26} T^{4}$$
5$D_{4}$ $$1 - 7666 T + 47007571 p^{2} T^{2} - 7666 p^{13} T^{3} + p^{26} T^{4}$$
7$D_{4}$ $$1 - 637048 T + 6020286906 p^{2} T^{2} - 637048 p^{13} T^{3} + p^{26} T^{4}$$
13$D_{4}$ $$1 + 291436 p T + 594138601660538 T^{2} + 291436 p^{14} T^{3} + p^{26} T^{4}$$
17$D_{4}$ $$1 - 37137304 T + 19126613212775662 T^{2} - 37137304 p^{13} T^{3} + p^{26} T^{4}$$
19$D_{4}$ $$1 + 102460596 T + 75240624868816198 T^{2} + 102460596 p^{13} T^{3} + p^{26} T^{4}$$
23$D_{4}$ $$1 - 1352747042 T + 1373469205060873963 T^{2} - 1352747042 p^{13} T^{3} + p^{26} T^{4}$$
29$D_{4}$ $$1 - 7425318120 T + 29655701354382887578 T^{2} - 7425318120 p^{13} T^{3} + p^{26} T^{4}$$
31$D_{4}$ $$1 - 8163482594 T + 57290834624075598491 T^{2} - 8163482594 p^{13} T^{3} + p^{26} T^{4}$$
37$D_{4}$ $$1 + 12073195594 T + 98252642021063024003 T^{2} + 12073195594 p^{13} T^{3} + p^{26} T^{4}$$
41$D_{4}$ $$1 - 73792259580 T +$$$$31\!\cdots\!18$$$$T^{2} - 73792259580 p^{13} T^{3} + p^{26} T^{4}$$
43$D_{4}$ $$1 - 20450919684 T +$$$$12\!\cdots\!46$$$$T^{2} - 20450919684 p^{13} T^{3} + p^{26} T^{4}$$
47$D_{4}$ $$1 - 71306154600 T +$$$$56\!\cdots\!50$$$$T^{2} - 71306154600 p^{13} T^{3} + p^{26} T^{4}$$
53$D_{4}$ $$1 - 309577967404 T +$$$$63\!\cdots\!46$$$$T^{2} - 309577967404 p^{13} T^{3} + p^{26} T^{4}$$
59$D_{4}$ $$1 + 403802069082 T +$$$$23\!\cdots\!43$$$$T^{2} + 403802069082 p^{13} T^{3} + p^{26} T^{4}$$
61$D_{4}$ $$1 + 81219577008 T +$$$$18\!\cdots\!82$$$$T^{2} + 81219577008 p^{13} T^{3} + p^{26} T^{4}$$
67$D_{4}$ $$1 - 229155633102 T +$$$$10\!\cdots\!31$$$$T^{2} - 229155633102 p^{13} T^{3} + p^{26} T^{4}$$
71$D_{4}$ $$1 + 1161878914578 T +$$$$15\!\cdots\!67$$$$T^{2} + 1161878914578 p^{13} T^{3} + p^{26} T^{4}$$
73$D_{4}$ $$1 - 456037317380 T -$$$$20\!\cdots\!50$$$$T^{2} - 456037317380 p^{13} T^{3} + p^{26} T^{4}$$
79$D_{4}$ $$1 - 63811251988 p T +$$$$15\!\cdots\!98$$$$T^{2} - 63811251988 p^{14} T^{3} + p^{26} T^{4}$$
83$D_{4}$ $$1 - 5610906244940 T +$$$$22\!\cdots\!30$$$$T^{2} - 5610906244940 p^{13} T^{3} + p^{26} T^{4}$$
89$D_{4}$ $$1 + 3239330626042 T +$$$$31\!\cdots\!23$$$$T^{2} + 3239330626042 p^{13} T^{3} + p^{26} T^{4}$$
97$D_{4}$ $$1 + 20541366120174 T +$$$$23\!\cdots\!23$$$$T^{2} + 20541366120174 p^{13} T^{3} + p^{26} T^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$