# Properties

 Label 2-138-1.1-c7-0-14 Degree $2$ Conductor $138$ Sign $1$ Analytic cond. $43.1091$ Root an. cond. $6.56575$ Motivic weight $7$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 8·2-s + 27·3-s + 64·4-s − 10.0·5-s + 216·6-s + 971.·7-s + 512·8-s + 729·9-s − 80.5·10-s − 2.06e3·11-s + 1.72e3·12-s + 9.53e3·13-s + 7.76e3·14-s − 271.·15-s + 4.09e3·16-s − 2.08e4·17-s + 5.83e3·18-s + 4.10e4·19-s − 644.·20-s + 2.62e4·21-s − 1.65e4·22-s − 1.21e4·23-s + 1.38e4·24-s − 7.80e4·25-s + 7.62e4·26-s + 1.96e4·27-s + 6.21e4·28-s + ⋯
 L(s)  = 1 + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.0360·5-s + 0.408·6-s + 1.07·7-s + 0.353·8-s + 0.333·9-s − 0.0254·10-s − 0.468·11-s + 0.288·12-s + 1.20·13-s + 0.756·14-s − 0.0207·15-s + 0.250·16-s − 1.02·17-s + 0.235·18-s + 1.37·19-s − 0.0180·20-s + 0.617·21-s − 0.331·22-s − 0.208·23-s + 0.204·24-s − 0.998·25-s + 0.851·26-s + 0.192·27-s + 0.535·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$138$$    =    $$2 \cdot 3 \cdot 23$$ Sign: $1$ Analytic conductor: $$43.1091$$ Root analytic conductor: $$6.56575$$ Motivic weight: $$7$$ Rational: no Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 138,\ (\ :7/2),\ 1)$$

## Particular Values

 $$L(4)$$ $$\approx$$ $$4.782560852$$ $$L(\frac12)$$ $$\approx$$ $$4.782560852$$ $$L(\frac{9}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 - 8T$$
3 $$1 - 27T$$
23 $$1 + 1.21e4T$$
good5 $$1 + 10.0T + 7.81e4T^{2}$$
7 $$1 - 971.T + 8.23e5T^{2}$$
11 $$1 + 2.06e3T + 1.94e7T^{2}$$
13 $$1 - 9.53e3T + 6.27e7T^{2}$$
17 $$1 + 2.08e4T + 4.10e8T^{2}$$
19 $$1 - 4.10e4T + 8.93e8T^{2}$$
29 $$1 - 2.46e4T + 1.72e10T^{2}$$
31 $$1 - 2.44e5T + 2.75e10T^{2}$$
37 $$1 - 5.20e5T + 9.49e10T^{2}$$
41 $$1 - 1.23e5T + 1.94e11T^{2}$$
43 $$1 - 7.42e3T + 2.71e11T^{2}$$
47 $$1 + 1.38e6T + 5.06e11T^{2}$$
53 $$1 - 1.01e6T + 1.17e12T^{2}$$
59 $$1 - 2.93e6T + 2.48e12T^{2}$$
61 $$1 + 2.36e6T + 3.14e12T^{2}$$
67 $$1 - 3.19e6T + 6.06e12T^{2}$$
71 $$1 + 3.49e6T + 9.09e12T^{2}$$
73 $$1 - 2.48e6T + 1.10e13T^{2}$$
79 $$1 + 4.80e6T + 1.92e13T^{2}$$
83 $$1 - 7.77e6T + 2.71e13T^{2}$$
89 $$1 - 5.39e5T + 4.42e13T^{2}$$
97 $$1 + 1.04e7T + 8.07e13T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$