# Properties

 Label 4-3e2-1.1-c13e2-0-0 Degree $4$ Conductor $9$ Sign $1$ Analytic cond. $10.3486$ Root an. cond. $1.79357$ Motivic weight $13$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 54·2-s + 1.45e3·3-s + 3.52e3·4-s + 4.07e4·5-s − 7.87e4·6-s − 2.10e4·7-s − 6.65e5·8-s + 1.59e6·9-s − 2.19e6·10-s + 6.72e5·11-s + 5.13e6·12-s + 1.75e7·13-s + 1.13e6·14-s + 5.93e7·15-s − 5.14e6·16-s + 8.38e7·17-s − 8.60e7·18-s + 2.56e8·19-s + 1.43e8·20-s − 3.06e7·21-s − 3.63e7·22-s + 8.59e8·23-s − 9.70e8·24-s − 9.07e8·25-s − 9.46e8·26-s + 1.54e9·27-s − 7.40e7·28-s + ⋯
 L(s)  = 1 − 0.596·2-s + 1.15·3-s + 0.430·4-s + 1.16·5-s − 0.688·6-s − 0.0674·7-s − 0.897·8-s + 9-s − 0.695·10-s + 0.114·11-s + 0.496·12-s + 1.00·13-s + 0.0402·14-s + 1.34·15-s − 0.0766·16-s + 0.842·17-s − 0.596·18-s + 1.24·19-s + 0.501·20-s − 0.0779·21-s − 0.0682·22-s + 1.21·23-s − 1.03·24-s − 0.743·25-s − 0.601·26-s + 0.769·27-s − 0.0290·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+13/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$9$$    =    $$3^{2}$$ Sign: $1$ Analytic conductor: $$10.3486$$ Root analytic conductor: $$1.79357$$ Motivic weight: $$13$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 9,\ (\ :13/2, 13/2),\ 1)$$

## Particular Values

 $$L(7)$$ $$\approx$$ $$2.320621641$$ $$L(\frac12)$$ $$\approx$$ $$2.320621641$$ $$L(\frac{15}{2})$$ 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)$
bad3$C_1$ $$( 1 - p^{6} T )^{2}$$
good2$D_{4}$ $$1 + 27 p T - 19 p^{5} T^{2} + 27 p^{14} T^{3} + p^{26} T^{4}$$
5$D_{4}$ $$1 - 40716 T + 102620542 p^{2} T^{2} - 40716 p^{13} T^{3} + p^{26} T^{4}$$
7$D_{4}$ $$1 + 21008 T - 2538590718 p T^{2} + 21008 p^{13} T^{3} + p^{26} T^{4}$$
11$D_{4}$ $$1 - 61128 p T + 375287075254 p^{2} T^{2} - 61128 p^{14} T^{3} + p^{26} T^{4}$$
13$D_{4}$ $$1 - 17532604 T + 539517124797054 T^{2} - 17532604 p^{13} T^{3} + p^{26} T^{4}$$
17$D_{4}$ $$1 - 83838564 T + 21472050522493798 T^{2} - 83838564 p^{13} T^{3} + p^{26} T^{4}$$
19$D_{4}$ $$1 - 256293544 T + 90096185084470998 T^{2} - 256293544 p^{13} T^{3} + p^{26} T^{4}$$
23$D_{4}$ $$1 - 859581936 T + 950532293946211246 T^{2} - 859581936 p^{13} T^{3} + p^{26} T^{4}$$
29$D_{4}$ $$1 + 4728475332 T + 25449630283285666078 T^{2} + 4728475332 p^{13} T^{3} + p^{26} T^{4}$$
31$D_{4}$ $$1 + 5982551648 T + 36024474609054776382 T^{2} + 5982551648 p^{13} T^{3} + p^{26} T^{4}$$
37$D_{4}$ $$1 - 27411194092 T +$$$$66\!\cdots\!74$$$$T^{2} - 27411194092 p^{13} T^{3} + p^{26} T^{4}$$
41$D_{4}$ $$1 - 15258974292 T +$$$$17\!\cdots\!82$$$$T^{2} - 15258974292 p^{13} T^{3} + p^{26} T^{4}$$
43$D_{4}$ $$1 + 11314499240 T +$$$$17\!\cdots\!50$$$$T^{2} + 11314499240 p^{13} T^{3} + p^{26} T^{4}$$
47$D_{4}$ $$1 + 69035142240 T +$$$$36\!\cdots\!10$$$$T^{2} + 69035142240 p^{13} T^{3} + p^{26} T^{4}$$
53$D_{4}$ $$1 + 226336894164 T +$$$$34\!\cdots\!66$$$$T^{2} + 226336894164 p^{13} T^{3} + p^{26} T^{4}$$
59$D_{4}$ $$1 + 927820824264 T +$$$$42\!\cdots\!38$$$$T^{2} + 927820824264 p^{13} T^{3} + p^{26} T^{4}$$
61$D_{4}$ $$1 - 179395461340 T +$$$$28\!\cdots\!98$$$$T^{2} - 179395461340 p^{13} T^{3} + p^{26} T^{4}$$
67$D_{4}$ $$1 + 698315061176 T +$$$$56\!\cdots\!18$$$$T^{2} + 698315061176 p^{13} T^{3} + p^{26} T^{4}$$
71$D_{4}$ $$1 + 784458549936 T +$$$$20\!\cdots\!46$$$$T^{2} + 784458549936 p^{13} T^{3} + p^{26} T^{4}$$
73$D_{4}$ $$1 - 1857400245076 T +$$$$33\!\cdots\!66$$$$T^{2} - 1857400245076 p^{13} T^{3} + p^{26} T^{4}$$
79$D_{4}$ $$1 + 714025470080 T +$$$$94\!\cdots\!78$$$$T^{2} + 714025470080 p^{13} T^{3} + p^{26} T^{4}$$
83$D_{4}$ $$1 + 4574293917912 T +$$$$22\!\cdots\!18$$$$T^{2} + 4574293917912 p^{13} T^{3} + p^{26} T^{4}$$
89$D_{4}$ $$1 - 3270178701684 T +$$$$37\!\cdots\!58$$$$T^{2} - 3270178701684 p^{13} T^{3} + p^{26} T^{4}$$
97$D_{4}$ $$1 + 9874926156476 T +$$$$13\!\cdots\!98$$$$T^{2} + 9874926156476 p^{13} T^{3} + p^{26} T^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$