# Properties

 Degree 2 Conductor $2^{2}$ Sign $1$ Motivic weight 4 Primitive yes Self-dual yes Analytic rank 0

# Origins

## Dirichlet series

 L(s)  = 1 − 4·2-s + 16·4-s − 14·5-s − 64·8-s + 81·9-s + 56·10-s − 238·13-s + 256·16-s + 322·17-s − 324·18-s − 224·20-s − 429·25-s + 952·26-s + 82·29-s − 1.02e3·32-s − 1.28e3·34-s + 1.29e3·36-s + 2.16e3·37-s + 896·40-s − 3.03e3·41-s − 1.13e3·45-s + 2.40e3·49-s + 1.71e3·50-s − 3.80e3·52-s + 2.48e3·53-s − 328·58-s − 6.95e3·61-s + ⋯
 L(s)  = 1 − 2-s + 4-s − 0.559·5-s − 8-s + 9-s + 0.559·10-s − 1.40·13-s + 16-s + 1.11·17-s − 18-s − 0.559·20-s − 0.686·25-s + 1.40·26-s + 0.0975·29-s − 32-s − 1.11·34-s + 36-s + 1.57·37-s + 0.559·40-s − 1.80·41-s − 0.559·45-s + 49-s + 0.686·50-s − 1.40·52-s + 0.883·53-s − 0.0975·58-s − 1.86·61-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$4$$    =    $$2^{2}$$ $$\varepsilon$$ = $1$ motivic weight = $$4$$ character : $\chi_{4} (3, \cdot )$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(2,\ 4,\ (\ :2),\ 1)$ $L(\frac{5}{2})$ $\approx$ $0.520074$ $L(\frac12)$ $\approx$ $0.520074$ $L(3)$ 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 2. If $p = 2$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 $$1 + p^{2} T$$
good3 $$( 1 - p^{2} T )( 1 + p^{2} T )$$
5 $$1 + 14 T + p^{4} T^{2}$$
7 $$( 1 - p^{2} T )( 1 + p^{2} T )$$
11 $$( 1 - p^{2} T )( 1 + p^{2} T )$$
13 $$1 + 238 T + p^{4} T^{2}$$
17 $$1 - 322 T + p^{4} T^{2}$$
19 $$( 1 - p^{2} T )( 1 + p^{2} T )$$
23 $$( 1 - p^{2} T )( 1 + p^{2} T )$$
29 $$1 - 82 T + p^{4} T^{2}$$
31 $$( 1 - p^{2} T )( 1 + p^{2} T )$$
37 $$1 - 2162 T + p^{4} T^{2}$$
41 $$1 + 3038 T + p^{4} T^{2}$$
43 $$( 1 - p^{2} T )( 1 + p^{2} T )$$
47 $$( 1 - p^{2} T )( 1 + p^{2} T )$$
53 $$1 - 2482 T + p^{4} T^{2}$$
59 $$( 1 - p^{2} T )( 1 + p^{2} T )$$
61 $$1 + 6958 T + p^{4} T^{2}$$
67 $$( 1 - p^{2} T )( 1 + p^{2} T )$$
71 $$( 1 - p^{2} T )( 1 + p^{2} T )$$
73 $$1 - 1442 T + p^{4} T^{2}$$
79 $$( 1 - p^{2} T )( 1 + p^{2} T )$$
83 $$( 1 - p^{2} T )( 1 + p^{2} T )$$
89 $$1 + 9758 T + p^{4} T^{2}$$
97 $$1 + 1918 T + p^{4} T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}