# Properties

 Degree $2$ Conductor $4410$ Sign $-1$ Motivic weight $1$ Primitive yes Self-dual yes Analytic rank $1$

# Learn more about

## Dirichlet series

 L(s)  = 1 − 2-s + 4-s − 5-s − 8-s + 10-s + 2·11-s + 16-s − 4·17-s + 6·19-s − 20-s − 2·22-s − 3·23-s + 25-s − 9·29-s + 4·31-s − 32-s + 4·34-s − 4·37-s − 6·38-s + 40-s − 7·41-s − 5·43-s + 2·44-s + 3·46-s + 8·47-s − 50-s + 2·53-s + ⋯
 L(s)  = 1 − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s + 0.316·10-s + 0.603·11-s + 1/4·16-s − 0.970·17-s + 1.37·19-s − 0.223·20-s − 0.426·22-s − 0.625·23-s + 1/5·25-s − 1.67·29-s + 0.718·31-s − 0.176·32-s + 0.685·34-s − 0.657·37-s − 0.973·38-s + 0.158·40-s − 1.09·41-s − 0.762·43-s + 0.301·44-s + 0.442·46-s + 1.16·47-s − 0.141·50-s + 0.274·53-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$4410$$    =    $$2 \cdot 3^{2} \cdot 5 \cdot 7^{2}$$ Sign: $-1$ Motivic weight: $$1$$ Character: $\chi_{4410} (1, \cdot )$ Sato-Tate group: $\mathrm{SU}(2)$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 4410,\ (\ :1/2),\ -1)$$

## Particular Values

 $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + T$$
3 $$1$$
5 $$1 + T$$
7 $$1$$
good11 $$1 - 2 T + p T^{2}$$
13 $$1 + p T^{2}$$
17 $$1 + 4 T + p T^{2}$$
19 $$1 - 6 T + p T^{2}$$
23 $$1 + 3 T + p T^{2}$$
29 $$1 + 9 T + p T^{2}$$
31 $$1 - 4 T + p T^{2}$$
37 $$1 + 4 T + p T^{2}$$
41 $$1 + 7 T + p T^{2}$$
43 $$1 + 5 T + p T^{2}$$
47 $$1 - 8 T + p T^{2}$$
53 $$1 - 2 T + p T^{2}$$
59 $$1 - 10 T + p T^{2}$$
61 $$1 + T + p T^{2}$$
67 $$1 + 9 T + p T^{2}$$
71 $$1 + 2 T + p T^{2}$$
73 $$1 - 4 T + p T^{2}$$
79 $$1 - 10 T + p T^{2}$$
83 $$1 + 7 T + p T^{2}$$
89 $$1 - T + p T^{2}$$
97 $$1 + 14 T + p T^{2}$$
show more
show less
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$

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

−18.39344793906594, −17.58391180736369, −17.16092809285176, −16.37401179894708, −15.99405355728317, −15.20219819896770, −14.87766900351538, −13.82772336681295, −13.48766231248538, −12.47041899344664, −11.82411437893010, −11.45785216715361, −10.71768237532792, −9.964599095735464, −9.348704359139376, −8.725416935142804, −8.065939793239334, −7.268949217236004, −6.817442448579542, −5.895888646202317, −5.095656951394345, −4.050840365928345, −3.371744555715663, −2.271797674369233, −1.282652289990869, 0, 1.282652289990869, 2.271797674369233, 3.371744555715663, 4.050840365928345, 5.095656951394345, 5.895888646202317, 6.817442448579542, 7.268949217236004, 8.065939793239334, 8.725416935142804, 9.348704359139376, 9.964599095735464, 10.71768237532792, 11.45785216715361, 11.82411437893010, 12.47041899344664, 13.48766231248538, 13.82772336681295, 14.87766900351538, 15.20219819896770, 15.99405355728317, 16.37401179894708, 17.16092809285176, 17.58391180736369, 18.39344793906594