# Properties

 Degree 2 Conductor $2^{2}$ Sign $1$ Motivic weight 32 Primitive yes Self-dual yes Analytic rank 0

# Origins

## Dirichlet series

 L(s)  = 1 + 6.55e4·2-s + 4.29e9·4-s − 1.96e11·5-s + 2.81e14·8-s + 1.85e15·9-s − 1.28e16·10-s + 1.33e18·13-s + 1.84e19·16-s + 1.42e18·17-s + 1.21e20·18-s − 8.43e20·20-s + 1.53e22·25-s + 8.71e22·26-s + 4.62e23·29-s + 1.20e24·32-s + 9.35e22·34-s + 7.95e24·36-s + 1.33e25·37-s − 5.53e25·40-s − 1.17e26·41-s − 3.64e26·45-s + 1.10e27·49-s + 1.00e27·50-s + 5.71e27·52-s − 6.73e27·53-s + 3.03e28·58-s − 7.13e28·61-s + ⋯
 L(s)  = 1 + 2-s + 4-s − 1.28·5-s + 8-s + 9-s − 1.28·10-s + 1.99·13-s + 16-s + 0.0293·17-s + 18-s − 1.28·20-s + 0.658·25-s + 1.99·26-s + 1.84·29-s + 32-s + 0.0293·34-s + 36-s + 1.08·37-s − 1.28·40-s − 1.84·41-s − 1.28·45-s + 49-s + 0.658·50-s + 1.99·52-s − 1.73·53-s + 1.84·58-s − 1.94·61-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(33-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+16) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$4$$    =    $$2^{2}$$ $$\varepsilon$$ = $1$ motivic weight = $$32$$ character : $\chi_{4} (3, \cdot )$ primitive : yes self-dual : yes analytic rank = $$0$$ Selberg data = $$(2,\ 4,\ (\ :16),\ 1)$$ $$L(\frac{33}{2})$$ $$\approx$$ $$3.642426946$$ $$L(\frac12)$$ $$\approx$$ $$3.642426946$$ $$L(17)$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \neq 2$,$$F_p(T)$$ is a polynomial of degree 2. If $p = 2$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1 - p^{16} T$$
good3 $$( 1 - p^{16} T )( 1 + p^{16} T )$$
5 $$1 + 196496109694 T + p^{32} T^{2}$$
7 $$( 1 - p^{16} T )( 1 + p^{16} T )$$
11 $$( 1 - p^{16} T )( 1 + p^{16} T )$$
13 $$1 - 1330087744899070082 T + p^{32} T^{2}$$
17 $$1 - 1427124567881986562 T + p^{32} T^{2}$$
19 $$( 1 - p^{16} T )( 1 + p^{16} T )$$
23 $$( 1 - p^{16} T )( 1 + p^{16} T )$$
29 $$1 -$$$$46\!\cdots\!42$$$$T + p^{32} T^{2}$$
31 $$( 1 - p^{16} T )( 1 + p^{16} T )$$
37 $$1 -$$$$13\!\cdots\!82$$$$T + p^{32} T^{2}$$
41 $$1 +$$$$11\!\cdots\!18$$$$T + p^{32} T^{2}$$
43 $$( 1 - p^{16} T )( 1 + p^{16} T )$$
47 $$( 1 - p^{16} T )( 1 + p^{16} T )$$
53 $$1 +$$$$67\!\cdots\!58$$$$T + p^{32} T^{2}$$
59 $$( 1 - p^{16} T )( 1 + p^{16} T )$$
61 $$1 +$$$$71\!\cdots\!78$$$$T + p^{32} T^{2}$$
67 $$( 1 - p^{16} T )( 1 + p^{16} T )$$
71 $$( 1 - p^{16} T )( 1 + p^{16} T )$$
73 $$1 -$$$$60\!\cdots\!22$$$$T + p^{32} T^{2}$$
79 $$( 1 - p^{16} T )( 1 + p^{16} T )$$
83 $$( 1 - p^{16} T )( 1 + p^{16} T )$$
89 $$1 -$$$$17\!\cdots\!22$$$$T + p^{32} T^{2}$$
97 $$1 -$$$$84\!\cdots\!42$$$$T + p^{32} T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}