# Properties

 Label 2-966-1.1-c1-0-10 Degree $2$ Conductor $966$ Sign $1$ Analytic cond. $7.71354$ Root an. cond. $2.77732$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $0$

# Origins

## Dirichlet series

 L(s)  = 1 + 2-s + 3-s + 4-s − 3·5-s + 6-s + 7-s + 8-s + 9-s − 3·10-s + 12-s + 5·13-s + 14-s − 3·15-s + 16-s + 18-s + 8·19-s − 3·20-s + 21-s − 23-s + 24-s + 4·25-s + 5·26-s + 27-s + 28-s + 3·29-s − 3·30-s + 2·31-s + ⋯
 L(s)  = 1 + 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.34·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.948·10-s + 0.288·12-s + 1.38·13-s + 0.267·14-s − 0.774·15-s + 1/4·16-s + 0.235·18-s + 1.83·19-s − 0.670·20-s + 0.218·21-s − 0.208·23-s + 0.204·24-s + 4/5·25-s + 0.980·26-s + 0.192·27-s + 0.188·28-s + 0.557·29-s − 0.547·30-s + 0.359·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$966$$    =    $$2 \cdot 3 \cdot 7 \cdot 23$$ Sign: $1$ Analytic conductor: $$7.71354$$ Root analytic conductor: $$2.77732$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 966,\ (\ :1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$2.711946717$$ $$L(\frac12)$$ $$\approx$$ $$2.711946717$$ $$L(\frac{3}{2})$$ 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 - T$$
7 $$1 - T$$
23 $$1 + T$$
good5 $$1 + 3 T + p T^{2}$$
11 $$1 + p T^{2}$$
13 $$1 - 5 T + p T^{2}$$
17 $$1 + p T^{2}$$
19 $$1 - 8 T + p T^{2}$$
29 $$1 - 3 T + p T^{2}$$
31 $$1 - 2 T + p T^{2}$$
37 $$1 + 7 T + p T^{2}$$
41 $$1 - 9 T + p T^{2}$$
43 $$1 + T + p T^{2}$$
47 $$1 + 3 T + p T^{2}$$
53 $$1 + 12 T + p T^{2}$$
59 $$1 + 6 T + p T^{2}$$
61 $$1 - 14 T + p T^{2}$$
67 $$1 + 4 T + p T^{2}$$
71 $$1 - 6 T + p T^{2}$$
73 $$1 + 4 T + p T^{2}$$
79 $$1 + 16 T + p T^{2}$$
83 $$1 + 12 T + p T^{2}$$
89 $$1 - 6 T + p T^{2}$$
97 $$1 + T + p T^{2}$$
show less
$$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

−10.12813423158950898419883848287, −8.991684707920905018023792625287, −8.131786680487201417413175908246, −7.62560968616237418084955999897, −6.70824904091140281523334700145, −5.54946185514318942435812651945, −4.49079541464908938937186749353, −3.69153174113643045200655187861, −2.99954201739737037557138342563, −1.29244537484774471452369042408, 1.29244537484774471452369042408, 2.99954201739737037557138342563, 3.69153174113643045200655187861, 4.49079541464908938937186749353, 5.54946185514318942435812651945, 6.70824904091140281523334700145, 7.62560968616237418084955999897, 8.131786680487201417413175908246, 8.991684707920905018023792625287, 10.12813423158950898419883848287