# Properties

 Degree 2 Conductor $2 \cdot 3^{3} \cdot 5^{2} \cdot 7$ Sign $-1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 1

# Origins

## Dirichlet series

 L(s)  = 1 + 2-s + 4-s + 7-s + 8-s − 11-s − 3·13-s + 14-s + 16-s + 8·17-s − 3·19-s − 22-s − 6·23-s − 3·26-s + 28-s − 6·29-s − 4·31-s + 32-s + 8·34-s − 2·37-s − 3·38-s − 11·41-s − 43-s − 44-s − 6·46-s − 47-s + 49-s − 3·52-s + ⋯
 L(s)  = 1 + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s − 0.301·11-s − 0.832·13-s + 0.267·14-s + 1/4·16-s + 1.94·17-s − 0.688·19-s − 0.213·22-s − 1.25·23-s − 0.588·26-s + 0.188·28-s − 1.11·29-s − 0.718·31-s + 0.176·32-s + 1.37·34-s − 0.328·37-s − 0.486·38-s − 1.71·41-s − 0.152·43-s − 0.150·44-s − 0.884·46-s − 0.145·47-s + 1/7·49-s − 0.416·52-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$9450$$    =    $$2 \cdot 3^{3} \cdot 5^{2} \cdot 7$$ $$\varepsilon$$ = $-1$ motivic weight = $$1$$ character : $\chi_{9450} (1, \cdot )$ primitive : yes self-dual : yes analytic rank = 1 Selberg data = $(2,\ 9450,\ (\ :1/2),\ -1)$ $L(1)$ $=$ $0$ $L(\frac12)$ $=$ $0$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;3,\;5,\;7\}$, $F_p(T) = 1 - a_p T + p T^2 .$If $p \in \{2,\;3,\;5,\;7\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 $$1 - T$$
3 $$1$$
5 $$1$$
7 $$1 - T$$
good11 $$1 + T + p T^{2}$$
13 $$1 + 3 T + p T^{2}$$
17 $$1 - 8 T + p T^{2}$$
19 $$1 + 3 T + p T^{2}$$
23 $$1 + 6 T + p T^{2}$$
29 $$1 + 6 T + p T^{2}$$
31 $$1 + 4 T + p T^{2}$$
37 $$1 + 2 T + p T^{2}$$
41 $$1 + 11 T + p T^{2}$$
43 $$1 + T + p T^{2}$$
47 $$1 + T + p T^{2}$$
53 $$1 - T + p T^{2}$$
59 $$1 + 10 T + p T^{2}$$
61 $$1 - 4 T + p T^{2}$$
67 $$1 - 3 T + p T^{2}$$
71 $$1 + 8 T + p T^{2}$$
73 $$1 + 11 T + p T^{2}$$
79 $$1 - 4 T + p T^{2}$$
83 $$1 - 11 T + p T^{2}$$
89 $$1 + 7 T + p T^{2}$$
97 $$1 + 2 T + p T^{2}$$
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\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}

## Imaginary part of the first few zeros on the critical line

−7.50497819389855751237713837020, −6.59788487551126754548506984690, −5.77681567302171082156520036288, −5.33258509131763330613282269062, −4.65379605198411918849061820626, −3.77897272297251269969679810219, −3.21272968600666125911987979012, −2.19652260487469449650208200845, −1.51864247643044039261478452422, 0, 1.51864247643044039261478452422, 2.19652260487469449650208200845, 3.21272968600666125911987979012, 3.77897272297251269969679810219, 4.65379605198411918849061820626, 5.33258509131763330613282269062, 5.77681567302171082156520036288, 6.59788487551126754548506984690, 7.50497819389855751237713837020