# Properties

 Label 4-3e2-1.1-c8e2-0-0 Degree $4$ Conductor $9$ Sign $1$ Analytic cond. $1.49361$ Root an. cond. $1.10550$ Motivic weight $8$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 90·3-s + 8·4-s − 3.50e3·7-s + 1.53e3·9-s + 720·12-s + 5.14e4·13-s − 6.54e4·16-s + 3.78e4·19-s − 3.15e5·21-s + 7.30e5·25-s − 4.51e5·27-s − 2.80e4·28-s − 7.02e5·31-s + 1.23e4·36-s + 2.67e6·37-s + 4.63e6·39-s − 7.05e6·43-s − 5.89e6·48-s − 2.34e6·49-s + 4.11e5·52-s + 3.40e6·57-s + 1.50e6·61-s − 5.38e6·63-s − 1.04e6·64-s + 4.53e6·67-s + 5.53e7·73-s + 6.57e7·75-s + ⋯
 L(s)  = 1 + 10/9·3-s + 1/32·4-s − 1.45·7-s + 0.234·9-s + 0.0347·12-s + 1.80·13-s − 0.999·16-s + 0.290·19-s − 1.61·21-s + 1.87·25-s − 0.850·27-s − 0.0455·28-s − 0.761·31-s + 0.00733·36-s + 1.42·37-s + 2.00·39-s − 2.06·43-s − 1.11·48-s − 0.406·49-s + 0.0563·52-s + 0.322·57-s + 0.108·61-s − 0.341·63-s − 0.0624·64-s + 0.225·67-s + 1.94·73-s + 2.07·75-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+4)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$9$$    =    $$3^{2}$$ Sign: $1$ Analytic conductor: $$1.49361$$ Root analytic conductor: $$1.10550$$ Motivic weight: $$8$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{3} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 9,\ (\ :4, 4),\ 1)$$

## Particular Values

 $$L(\frac{9}{2})$$ $$\approx$$ $$1.382913471$$ $$L(\frac12)$$ $$\approx$$ $$1.382913471$$ $$L(5)$$ 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$C_2$ $$1 - 10 p^{2} T + p^{8} T^{2}$$
good2$C_2^2$ $$1 - p^{3} T^{2} + p^{16} T^{4}$$
5$C_2^2$ $$1 - 29234 p^{2} T^{2} + p^{16} T^{4}$$
7$C_2$ $$( 1 + 250 p T + p^{8} T^{2} )^{2}$$
11$C_2^2$ $$1 - 380283362 T^{2} + p^{16} T^{4}$$
13$C_2$ $$( 1 - 25730 T + p^{8} T^{2} )^{2}$$
17$C_2^2$ $$1 - 8342551298 T^{2} + p^{16} T^{4}$$
19$C_2$ $$( 1 - 18938 T + p^{8} T^{2} )^{2}$$
23$C_2^2$ $$1 + 64711613182 T^{2} + p^{16} T^{4}$$
29$C_2^2$ $$1 - 788066452322 T^{2} + p^{16} T^{4}$$
31$C_2$ $$( 1 + 11338 p T + p^{8} T^{2} )^{2}$$
37$C_2$ $$( 1 - 1335170 T + p^{8} T^{2} )^{2}$$
41$C_2^2$ $$1 - 12452468931842 T^{2} + p^{16} T^{4}$$
43$C_2$ $$( 1 + 3526150 T + p^{8} T^{2} )^{2}$$
47$C_2^2$ $$1 - 30967680304898 T^{2} + p^{16} T^{4}$$
53$C_2^2$ $$1 - 80936075395298 T^{2} + p^{16} T^{4}$$
59$C_2^2$ $$1 - 105562517046242 T^{2} + p^{16} T^{4}$$
61$C_2$ $$( 1 - 753602 T + p^{8} T^{2} )^{2}$$
67$C_2$ $$( 1 - 2268890 T + p^{8} T^{2} )^{2}$$
71$C_2^2$ $$1 - 1001758688017922 T^{2} + p^{16} T^{4}$$
73$C_2$ $$( 1 - 27672770 T + p^{8} T^{2} )^{2}$$
79$C_2$ $$( 1 + 22980982 T + p^{8} T^{2} )^{2}$$
83$C_2^2$ $$1 - 2352070843223138 T^{2} + p^{16} T^{4}$$
89$C_2^2$ $$1 - 2600204109557762 T^{2} + p^{16} T^{4}$$
97$C_2$ $$( 1 - 147271010 T + p^{8} T^{2} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$