# 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 0

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## Dirichlet series

 L(s)  = 1 − 2-s + 4-s + 7-s − 8-s − 2·11-s + 5·13-s − 14-s + 16-s + 6·17-s + 6·19-s + 2·22-s − 8·23-s − 5·26-s + 28-s + 5·29-s + 7·31-s − 32-s − 6·34-s + 4·37-s − 6·38-s − 41-s + 4·43-s − 2·44-s + 8·46-s − 2·47-s + 49-s + 5·52-s + ⋯
 L(s)  = 1 − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s − 0.603·11-s + 1.38·13-s − 0.267·14-s + 1/4·16-s + 1.45·17-s + 1.37·19-s + 0.426·22-s − 1.66·23-s − 0.980·26-s + 0.188·28-s + 0.928·29-s + 1.25·31-s − 0.176·32-s − 1.02·34-s + 0.657·37-s − 0.973·38-s − 0.156·41-s + 0.609·43-s − 0.301·44-s + 1.17·46-s − 0.291·47-s + 1/7·49-s + 0.693·52-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 = 0 Selberg data = $(2,\ 9450,\ (\ :1/2),\ 1)$ $L(1)$ $\approx$ $1.885603744$ $L(\frac12)$ $\approx$ $1.885603744$ $L(\frac{3}{2})$ 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 + 2 T + p T^{2}$$
13 $$1 - 5 T + p T^{2}$$
17 $$1 - 6 T + p T^{2}$$
19 $$1 - 6 T + p T^{2}$$
23 $$1 + 8 T + p T^{2}$$
29 $$1 - 5 T + p T^{2}$$
31 $$1 - 7 T + p T^{2}$$
37 $$1 - 4 T + p T^{2}$$
41 $$1 + T + p T^{2}$$
43 $$1 - 4 T + p T^{2}$$
47 $$1 + 2 T + p T^{2}$$
53 $$1 - 13 T + p T^{2}$$
59 $$1 + 4 T + p T^{2}$$
61 $$1 + 9 T + p T^{2}$$
67 $$1 + 9 T + p T^{2}$$
71 $$1 - 15 T + p T^{2}$$
73 $$1 + 12 T + p T^{2}$$
79 $$1 - 10 T + p T^{2}$$
83 $$1 - 5 T + p T^{2}$$
89 $$1 + 11 T + p T^{2}$$
97 $$1 + 8 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

−7.955306972882156411046145235557, −7.25897008423325185889933050977, −6.22909204676621175697952521133, −5.84898447064486668557891821784, −5.08245920273153004138635153784, −4.09912831009139447668038383358, −3.30645760049096373522973532262, −2.57562071040025305858443625141, −1.44495762941669005269705758245, −0.814341757215904798544444927775, 0.814341757215904798544444927775, 1.44495762941669005269705758245, 2.57562071040025305858443625141, 3.30645760049096373522973532262, 4.09912831009139447668038383358, 5.08245920273153004138635153784, 5.84898447064486668557891821784, 6.22909204676621175697952521133, 7.25897008423325185889933050977, 7.955306972882156411046145235557