# Properties

 Degree 2 Conductor $2^{3} \cdot 3$ Sign $1$ Motivic weight 7 Primitive yes Self-dual yes Analytic rank 0

# Origins

## Dirichlet series

 L(s)  = 1 + 27·3-s + 110·5-s + 504·7-s + 729·9-s + 3.81e3·11-s + 9.57e3·13-s + 2.97e3·15-s + 2.60e4·17-s − 3.83e4·19-s + 1.36e4·21-s − 7.11e4·23-s − 6.60e4·25-s + 1.96e4·27-s + 7.42e4·29-s − 2.75e5·31-s + 1.02e5·33-s + 5.54e4·35-s − 2.66e5·37-s + 2.58e5·39-s + 6.84e5·41-s + 2.45e5·43-s + 8.01e4·45-s + 4.78e5·47-s − 5.69e5·49-s + 7.04e5·51-s − 5.69e5·53-s + 4.19e5·55-s + ⋯
 L(s)  = 1 + 0.577·3-s + 0.393·5-s + 0.555·7-s + 1/3·9-s + 0.863·11-s + 1.20·13-s + 0.227·15-s + 1.28·17-s − 1.28·19-s + 0.320·21-s − 1.21·23-s − 0.845·25-s + 0.192·27-s + 0.565·29-s − 1.66·31-s + 0.498·33-s + 0.218·35-s − 0.865·37-s + 0.697·39-s + 1.55·41-s + 0.471·43-s + 0.131·45-s + 0.672·47-s − 0.691·49-s + 0.743·51-s − 0.525·53-s + 0.339·55-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$24$$    =    $$2^{3} \cdot 3$$ $$\varepsilon$$ = $1$ motivic weight = $$7$$ character : $\chi_{24} (1, \cdot )$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(2,\ 24,\ (\ :7/2),\ 1)$ $L(4)$ $\approx$ $2.27801$ $L(\frac12)$ $\approx$ $2.27801$ $L(\frac{9}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;3\}$, $$F_p$$ is a polynomial of degree 2. If $p \in \{2,\;3\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 $$1$$
3 $$1 - p^{3} T$$
good5 $$1 - 22 p T + p^{7} T^{2}$$
7 $$1 - 72 p T + p^{7} T^{2}$$
11 $$1 - 3812 T + p^{7} T^{2}$$
13 $$1 - 9574 T + p^{7} T^{2}$$
17 $$1 - 26098 T + p^{7} T^{2}$$
19 $$1 + 38308 T + p^{7} T^{2}$$
23 $$1 + 71128 T + p^{7} T^{2}$$
29 $$1 - 74262 T + p^{7} T^{2}$$
31 $$1 + 275680 T + p^{7} T^{2}$$
37 $$1 + 266610 T + p^{7} T^{2}$$
41 $$1 - 684762 T + p^{7} T^{2}$$
43 $$1 - 245956 T + p^{7} T^{2}$$
47 $$1 - 478800 T + p^{7} T^{2}$$
53 $$1 + 569410 T + p^{7} T^{2}$$
59 $$1 + 1525324 T + p^{7} T^{2}$$
61 $$1 + 2640458 T + p^{7} T^{2}$$
67 $$1 - 1416236 T + p^{7} T^{2}$$
71 $$1 + 3511304 T + p^{7} T^{2}$$
73 $$1 - 4738618 T + p^{7} T^{2}$$
79 $$1 - 4661488 T + p^{7} T^{2}$$
83 $$1 + 5729252 T + p^{7} T^{2}$$
89 $$1 - 11993514 T + p^{7} T^{2}$$
97 $$1 - 7150754 T + p^{7} T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}