# Properties

 Label 2-147-1.1-c1-0-4 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 $0$

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## Dirichlet series

 L(s)  = 1 + 2·2-s − 3-s + 2·4-s + 2·5-s − 2·6-s + 9-s + 4·10-s − 2·11-s − 2·12-s − 13-s − 2·15-s − 4·16-s + 2·18-s − 19-s + 4·20-s − 4·22-s − 25-s − 2·26-s − 27-s + 4·29-s − 4·30-s − 9·31-s − 8·32-s + 2·33-s + 2·36-s + 3·37-s − 2·38-s + ⋯
 L(s)  = 1 + 1.41·2-s − 0.577·3-s + 4-s + 0.894·5-s − 0.816·6-s + 1/3·9-s + 1.26·10-s − 0.603·11-s − 0.577·12-s − 0.277·13-s − 0.516·15-s − 16-s + 0.471·18-s − 0.229·19-s + 0.894·20-s − 0.852·22-s − 1/5·25-s − 0.392·26-s − 0.192·27-s + 0.742·29-s − 0.730·30-s − 1.61·31-s − 1.41·32-s + 0.348·33-s + 1/3·36-s + 0.493·37-s − 0.324·38-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

 $$L(1)$$ $$\approx$$ $$1.920053745$$ $$L(\frac12)$$ $$\approx$$ $$1.920053745$$ $$L(\frac{3}{2})$$ 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 - p T + p T^{2}$$
5 $$1 - 2 T + p T^{2}$$
11 $$1 + 2 T + p T^{2}$$
13 $$1 + T + p T^{2}$$
17 $$1 + p T^{2}$$
19 $$1 + T + p T^{2}$$
23 $$1 + p T^{2}$$
29 $$1 - 4 T + p T^{2}$$
31 $$1 + 9 T + p T^{2}$$
37 $$1 - 3 T + p T^{2}$$
41 $$1 - 10 T + p T^{2}$$
43 $$1 - 5 T + p T^{2}$$
47 $$1 - 6 T + p T^{2}$$
53 $$1 - 12 T + p T^{2}$$
59 $$1 - 12 T + p T^{2}$$
61 $$1 + 10 T + p T^{2}$$
67 $$1 + 5 T + p T^{2}$$
71 $$1 + 6 T + p T^{2}$$
73 $$1 - 3 T + p T^{2}$$
79 $$1 + T + p T^{2}$$
83 $$1 + 6 T + p T^{2}$$
89 $$1 + 16 T + p T^{2}$$
97 $$1 - 6 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

−13.05041447442840207720378708897, −12.40866641446903147774840263019, −11.30162828605416851531895428330, −10.28136035144373451283122510308, −9.092563983560312555964020990816, −7.29485403540511127428179965078, −6.01621490934601452163404663718, −5.40855511243246313730427714163, −4.18739539622885934722496613081, −2.47062022887811000287520417928, 2.47062022887811000287520417928, 4.18739539622885934722496613081, 5.40855511243246313730427714163, 6.01621490934601452163404663718, 7.29485403540511127428179965078, 9.092563983560312555964020990816, 10.28136035144373451283122510308, 11.30162828605416851531895428330, 12.40866641446903147774840263019, 13.05041447442840207720378708897