# Properties

 Label 2-9660-1.1-c1-0-73 Degree $2$ Conductor $9660$ Sign $-1$ Analytic cond. $77.1354$ Root an. cond. $8.78268$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $1$

# Origins

## Dirichlet series

 L(s)  = 1 − 3-s − 5-s + 7-s + 9-s + 4·11-s + 6·13-s + 15-s − 6·17-s + 6·19-s − 21-s − 23-s + 25-s − 27-s − 2·29-s − 8·31-s − 4·33-s − 35-s − 8·37-s − 6·39-s − 10·41-s − 4·43-s − 45-s − 12·47-s + 49-s + 6·51-s − 6·53-s − 4·55-s + ⋯
 L(s)  = 1 − 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s + 1.20·11-s + 1.66·13-s + 0.258·15-s − 1.45·17-s + 1.37·19-s − 0.218·21-s − 0.208·23-s + 1/5·25-s − 0.192·27-s − 0.371·29-s − 1.43·31-s − 0.696·33-s − 0.169·35-s − 1.31·37-s − 0.960·39-s − 1.56·41-s − 0.609·43-s − 0.149·45-s − 1.75·47-s + 1/7·49-s + 0.840·51-s − 0.824·53-s − 0.539·55-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 9660 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 9660 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$9660$$    =    $$2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 23$$ Sign: $-1$ Analytic conductor: $$77.1354$$ Root analytic conductor: $$8.78268$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: $\chi_{9660} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 9660,\ (\ :1/2),\ -1)$$

## 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$$
3 $$1 + T$$
5 $$1 + T$$
7 $$1 - T$$
23 $$1 + T$$
good11 $$1 - 4 T + p T^{2}$$
13 $$1 - 6 T + p T^{2}$$
17 $$1 + 6 T + p T^{2}$$
19 $$1 - 6 T + p T^{2}$$
29 $$1 + 2 T + p T^{2}$$
31 $$1 + 8 T + p T^{2}$$
37 $$1 + 8 T + p T^{2}$$
41 $$1 + 10 T + p T^{2}$$
43 $$1 + 4 T + p T^{2}$$
47 $$1 + 12 T + p T^{2}$$
53 $$1 + 6 T + p T^{2}$$
59 $$1 - 10 T + p T^{2}$$
61 $$1 - 6 T + p T^{2}$$
67 $$1 + 12 T + p T^{2}$$
71 $$1 + 2 T + p T^{2}$$
73 $$1 - 6 T + p T^{2}$$
79 $$1 + 2 T + p T^{2}$$
83 $$1 - 4 T + p T^{2}$$
89 $$1 + 14 T + p T^{2}$$
97 $$1 - 4 T + p T^{2}$$
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$$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

−7.03006976309984419732306100212, −6.80743205754775227254413357550, −6.01057369855440498797083645690, −5.28786113739241214909370411714, −4.58694293825222626034596344591, −3.69423974735149489940934354479, −3.42865933473498536313227527018, −1.81416745756692951175793153535, −1.30534759960674675816245604323, 0, 1.30534759960674675816245604323, 1.81416745756692951175793153535, 3.42865933473498536313227527018, 3.69423974735149489940934354479, 4.58694293825222626034596344591, 5.28786113739241214909370411714, 6.01057369855440498797083645690, 6.80743205754775227254413357550, 7.03006976309984419732306100212