# Properties

 Degree 2 Conductor $3 \cdot 5 \cdot 23 \cdot 29$ Sign $-1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 1

# Related objects

## Dirichlet series

 L(s)  = 1 − 3-s − 2·4-s − 5-s − 5·7-s + 9-s + 2·12-s − 4·13-s + 15-s + 4·16-s + 3·17-s − 7·19-s + 2·20-s + 5·21-s − 23-s + 25-s − 27-s + 10·28-s − 29-s − 8·31-s + 5·35-s − 2·36-s + 4·37-s + 4·39-s + 2·41-s + 8·43-s − 45-s + 6·47-s + ⋯
 L(s)  = 1 − 0.577·3-s − 4-s − 0.447·5-s − 1.88·7-s + 1/3·9-s + 0.577·12-s − 1.10·13-s + 0.258·15-s + 16-s + 0.727·17-s − 1.60·19-s + 0.447·20-s + 1.09·21-s − 0.208·23-s + 1/5·25-s − 0.192·27-s + 1.88·28-s − 0.185·29-s − 1.43·31-s + 0.845·35-s − 1/3·36-s + 0.657·37-s + 0.640·39-s + 0.312·41-s + 1.21·43-s − 0.149·45-s + 0.875·47-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 10005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 10005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$10005$$    =    $$3 \cdot 5 \cdot 23 \cdot 29$$ $$\varepsilon$$ = $-1$ motivic weight = $$1$$ character : $\chi_{10005} (1, \cdot )$ Sato-Tate : $\mathrm{SU}(2)$ primitive : yes self-dual : yes analytic rank = $$1$$ Selberg data = $$(2,\ 10005,\ (\ :1/2),\ -1)$$ $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{3,\;5,\;23,\;29\}$,$F_p(T) = 1 - a_p T + p T^2 .$If $p \in \{3,\;5,\;23,\;29\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 $$1 + T$$
5 $$1 + T$$
23 $$1 + T$$
29 $$1 + T$$
good2 $$1 + p T^{2}$$
7 $$1 + 5 T + p T^{2}$$
11 $$1 + p T^{2}$$
13 $$1 + 4 T + p T^{2}$$
17 $$1 - 3 T + p T^{2}$$
19 $$1 + 7 T + p T^{2}$$
31 $$1 + 8 T + p T^{2}$$
37 $$1 - 4 T + p T^{2}$$
41 $$1 - 2 T + p T^{2}$$
43 $$1 - 8 T + p T^{2}$$
47 $$1 - 6 T + p T^{2}$$
53 $$1 + 4 T + p T^{2}$$
59 $$1 + 10 T + p T^{2}$$
61 $$1 - 3 T + p T^{2}$$
67 $$1 + p T^{2}$$
71 $$1 - 8 T + p T^{2}$$
73 $$1 - 11 T + p T^{2}$$
79 $$1 - 9 T + p T^{2}$$
83 $$1 + 14 T + p T^{2}$$
89 $$1 - 8 T + p T^{2}$$
97 $$1 - 2 T + p T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}