Properties

 Label 2-3e2-1.1-c3-0-0 Degree $2$ Conductor $9$ Sign $1$ Analytic cond. $0.531017$ Root an. cond. $0.728709$ Motivic weight $3$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $0$

Learn more

Dirichlet series

 L(s)  = 1 − 8·4-s + 20·7-s − 70·13-s + 64·16-s + 56·19-s − 125·25-s − 160·28-s + 308·31-s + 110·37-s − 520·43-s + 57·49-s + 560·52-s + 182·61-s − 512·64-s − 880·67-s + 1.19e3·73-s − 448·76-s + 884·79-s − 1.40e3·91-s − 1.33e3·97-s + 1.00e3·100-s + 1.82e3·103-s − 646·109-s + 1.28e3·112-s + ⋯
 L(s)  = 1 − 4-s + 1.07·7-s − 1.49·13-s + 16-s + 0.676·19-s − 25-s − 1.07·28-s + 1.78·31-s + 0.488·37-s − 1.84·43-s + 0.166·49-s + 1.49·52-s + 0.382·61-s − 64-s − 1.60·67-s + 1.90·73-s − 0.676·76-s + 1.25·79-s − 1.61·91-s − 1.39·97-s + 100-s + 1.74·103-s − 0.567·109-s + 1.07·112-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$9$$    =    $$3^{2}$$ Sign: $1$ Analytic conductor: $$0.531017$$ Root analytic conductor: $$0.728709$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 9,\ (\ :3/2),\ 1)$$

Particular Values

 $$L(2)$$ $$\approx$$ $$0.7599657499$$ $$L(\frac12)$$ $$\approx$$ $$0.7599657499$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1$$
good2 $$1 + p^{3} T^{2}$$
5 $$1 + p^{3} T^{2}$$
7 $$1 - 20 T + p^{3} T^{2}$$
11 $$1 + p^{3} T^{2}$$
13 $$1 + 70 T + p^{3} T^{2}$$
17 $$1 + p^{3} T^{2}$$
19 $$1 - 56 T + p^{3} T^{2}$$
23 $$1 + p^{3} T^{2}$$
29 $$1 + p^{3} T^{2}$$
31 $$1 - 308 T + p^{3} T^{2}$$
37 $$1 - 110 T + p^{3} T^{2}$$
41 $$1 + p^{3} T^{2}$$
43 $$1 + 520 T + p^{3} T^{2}$$
47 $$1 + p^{3} T^{2}$$
53 $$1 + p^{3} T^{2}$$
59 $$1 + p^{3} T^{2}$$
61 $$1 - 182 T + p^{3} T^{2}$$
67 $$1 + 880 T + p^{3} T^{2}$$
71 $$1 + p^{3} T^{2}$$
73 $$1 - 1190 T + p^{3} T^{2}$$
79 $$1 - 884 T + p^{3} T^{2}$$
83 $$1 + p^{3} T^{2}$$
89 $$1 + p^{3} T^{2}$$
97 $$1 + 1330 T + p^{3} 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

−21.24661792052083109285108792256, −19.55575161404065204005050768737, −18.07795890634270969575136593189, −17.12497617568479204935496377782, −14.93321934083369061780725752962, −13.74456961426938286092483499795, −11.94315109526654413183623103861, −9.820987617456067807528942581344, −7.996008136155744588027072213229, −4.88396746546602162317969751078, 4.88396746546602162317969751078, 7.996008136155744588027072213229, 9.820987617456067807528942581344, 11.94315109526654413183623103861, 13.74456961426938286092483499795, 14.93321934083369061780725752962, 17.12497617568479204935496377782, 18.07795890634270969575136593189, 19.55575161404065204005050768737, 21.24661792052083109285108792256