Properties

 Label 2-147-1.1-c1-0-0 Degree $2$ Conductor $147$ Sign $1$ Analytic cond. $1.17380$ Root an. cond. $1.08342$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $0$

Related objects

Dirichlet series

 L(s)  = 1 − 2-s − 3-s − 4-s + 2·5-s + 6-s + 3·8-s + 9-s − 2·10-s + 4·11-s + 12-s + 2·13-s − 2·15-s − 16-s + 6·17-s − 18-s − 4·19-s − 2·20-s − 4·22-s − 3·24-s − 25-s − 2·26-s − 27-s − 2·29-s + 2·30-s − 5·32-s − 4·33-s − 6·34-s + ⋯
 L(s)  = 1 − 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.894·5-s + 0.408·6-s + 1.06·8-s + 1/3·9-s − 0.632·10-s + 1.20·11-s + 0.288·12-s + 0.554·13-s − 0.516·15-s − 1/4·16-s + 1.45·17-s − 0.235·18-s − 0.917·19-s − 0.447·20-s − 0.852·22-s − 0.612·24-s − 1/5·25-s − 0.392·26-s − 0.192·27-s − 0.371·29-s + 0.365·30-s − 0.883·32-s − 0.696·33-s − 1.02·34-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$147$$    =    $$3 \cdot 7^{2}$$ Sign: $1$ Analytic conductor: $$1.17380$$ Root analytic conductor: $$1.08342$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 147,\ (\ :1/2),\ 1)$$

Particular Values

 $$L(1)$$ $$\approx$$ $$0.7222862454$$ $$L(\frac12)$$ $$\approx$$ $$0.7222862454$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + T$$
7 $$1$$
good2 $$1 + T + p T^{2}$$
5 $$1 - 2 T + p T^{2}$$
11 $$1 - 4 T + p T^{2}$$
13 $$1 - 2 T + p T^{2}$$
17 $$1 - 6 T + p T^{2}$$
19 $$1 + 4 T + p T^{2}$$
23 $$1 + p T^{2}$$
29 $$1 + 2 T + p T^{2}$$
31 $$1 + p T^{2}$$
37 $$1 - 6 T + p T^{2}$$
41 $$1 + 2 T + p T^{2}$$
43 $$1 + 4 T + p T^{2}$$
47 $$1 + p T^{2}$$
53 $$1 - 6 T + p T^{2}$$
59 $$1 + 12 T + p T^{2}$$
61 $$1 - 2 T + p T^{2}$$
67 $$1 - 4 T + p T^{2}$$
71 $$1 + p T^{2}$$
73 $$1 - 6 T + p T^{2}$$
79 $$1 + 16 T + p T^{2}$$
83 $$1 - 12 T + p T^{2}$$
89 $$1 - 14 T + p T^{2}$$
97 $$1 + 18 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

−13.09697998821175856125894317781, −11.98659319295797721290412051260, −10.77789244508403227045944829665, −9.863214598051417092763914791206, −9.163045893798789171919999202805, −7.977686193886306753143677629185, −6.53484921474790582970192984243, −5.48510765590071063887274763802, −4.01671863219499060051377207353, −1.40786771277750203137263322761, 1.40786771277750203137263322761, 4.01671863219499060051377207353, 5.48510765590071063887274763802, 6.53484921474790582970192984243, 7.977686193886306753143677629185, 9.163045893798789171919999202805, 9.863214598051417092763914791206, 10.77789244508403227045944829665, 11.98659319295797721290412051260, 13.09697998821175856125894317781