# Properties

 Degree 4 Conductor $3^{4}$ Sign $1$ Motivic weight 11 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 1.57e3·4-s + 1.16e5·7-s + 1.52e6·13-s − 1.71e6·16-s − 2.06e7·19-s + 2.87e7·25-s − 1.83e8·28-s + 2.12e8·31-s − 1.91e7·37-s + 3.18e9·43-s + 6.17e9·49-s − 2.40e9·52-s − 6.18e9·61-s + 9.30e9·64-s − 1.82e10·67-s + 1.24e9·73-s + 3.24e10·76-s + 2.12e10·79-s + 1.77e11·91-s + 2.63e11·97-s − 4.53e10·100-s + 1.59e11·103-s − 5.78e10·109-s − 1.98e11·112-s − 5.44e11·121-s − 3.34e11·124-s + 127-s + ⋯
 L(s)  = 1 − 0.769·4-s + 2.61·7-s + 1.13·13-s − 0.407·16-s − 1.90·19-s + 0.589·25-s − 2.01·28-s + 1.33·31-s − 0.0453·37-s + 3.30·43-s + 3.12·49-s − 0.876·52-s − 0.937·61-s + 1.08·64-s − 1.64·67-s + 0.0700·73-s + 1.46·76-s + 0.776·79-s + 2.97·91-s + 3.11·97-s − 0.453·100-s + 1.35·103-s − 0.360·109-s − 1.06·112-s − 1.90·121-s − 1.02·124-s − 4.98·133-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+11/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$81$$    =    $$3^{4}$$ $$\varepsilon$$ = $1$ motivic weight = $$11$$ character : induced by $\chi_{9} (1, \cdot )$ primitive : no self-dual : yes analytic rank = 0 Selberg data = $(4,\ 81,\ (\ :11/2, 11/2),\ 1)$ $L(6)$ $\approx$ $2.41750$ $L(\frac12)$ $\approx$ $2.41750$ $L(\frac{13}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \neq 3$, $$F_p(T)$$ is a polynomial of degree 4. If $p = 3$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad3 $$1$$
good2$C_2^2$ $$1 + 197 p^{3} T^{2} + p^{22} T^{4}$$
5$C_2^2$ $$1 - 5757454 p T^{2} + p^{22} T^{4}$$
7$C_2$ $$( 1 - 8300 p T + p^{11} T^{2} )^{2}$$
11$C_2^2$ $$1 + 544818541222 T^{2} + p^{22} T^{4}$$
13$C_2$ $$( 1 - 762650 T + p^{11} T^{2} )^{2}$$
17$C_2^2$ $$1 - 11975946694814 T^{2} + p^{22} T^{4}$$
19$C_2$ $$( 1 + 10301704 T + p^{11} T^{2} )^{2}$$
23$C_2^2$ $$1 + 1797531507451534 T^{2} + p^{22} T^{4}$$
29$C_2^2$ $$1 + 4519838782611658 T^{2} + p^{22} T^{4}$$
31$C_2$ $$( 1 - 106159508 T + p^{11} T^{2} )^{2}$$
37$C_2$ $$( 1 + 9574450 T + p^{11} T^{2} )^{2}$$
41$C_2^2$ $$1 + 1088883326741296882 T^{2} + p^{22} T^{4}$$
43$C_2$ $$( 1 - 1590697400 T + p^{11} T^{2} )^{2}$$
47$C_2^2$ $$1 + 2869299998598432286 T^{2} + p^{22} T^{4}$$
53$C_2^2$ $$1 + 17432708152043665114 T^{2} + p^{22} T^{4}$$
59$C_2^2$ $$1 + 26931896409526885318 T^{2} + p^{22} T^{4}$$
61$C_2$ $$( 1 + 3092621098 T + p^{11} T^{2} )^{2}$$
67$C_2$ $$( 1 + 9113820400 T + p^{11} T^{2} )^{2}$$
71$C_2^2$ $$1 +$$$$45\!\cdots\!42$$$$T^{2} + p^{22} T^{4}$$
73$C_2$ $$( 1 - 620142950 T + p^{11} T^{2} )^{2}$$
79$C_2$ $$( 1 - 10618486484 T + p^{11} T^{2} )^{2}$$
83$C_2^2$ $$1 -$$$$10\!\cdots\!46$$$$T^{2} + p^{22} T^{4}$$
89$C_2^2$ $$1 +$$$$17\!\cdots\!78$$$$T^{2} + p^{22} T^{4}$$
97$C_2$ $$( 1 - 131872902350 T + p^{11} T^{2} )^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}