# Properties

 Label 2-1617-1.1-c3-0-181 Degree $2$ Conductor $1617$ Sign $1$ Analytic cond. $95.4060$ Root an. cond. $9.76760$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 5.42·2-s + 3·3-s + 21.4·4-s + 16.8·5-s + 16.2·6-s + 72.8·8-s + 9·9-s + 91.3·10-s − 11·11-s + 64.2·12-s − 24.8·13-s + 50.5·15-s + 223.·16-s + 15.9·17-s + 48.8·18-s − 15.1·19-s + 360.·20-s − 59.6·22-s + 17.7·23-s + 218.·24-s + 158.·25-s − 134.·26-s + 27·27-s − 128.·29-s + 274.·30-s − 219.·31-s + 630.·32-s + ⋯
 L(s)  = 1 + 1.91·2-s + 0.577·3-s + 2.67·4-s + 1.50·5-s + 1.10·6-s + 3.21·8-s + 0.333·9-s + 2.89·10-s − 0.301·11-s + 1.54·12-s − 0.530·13-s + 0.870·15-s + 3.49·16-s + 0.227·17-s + 0.639·18-s − 0.182·19-s + 4.03·20-s − 0.578·22-s + 0.160·23-s + 1.85·24-s + 1.27·25-s − 1.01·26-s + 0.192·27-s − 0.823·29-s + 1.66·30-s − 1.27·31-s + 3.48·32-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1617$$    =    $$3 \cdot 7^{2} \cdot 11$$ Sign: $1$ Analytic conductor: $$95.4060$$ Root analytic conductor: $$9.76760$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 1617,\ (\ :3/2),\ 1)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$13.29105286$$ $$L(\frac12)$$ $$\approx$$ $$13.29105286$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 - 3T$$
7 $$1$$
11 $$1 + 11T$$
good2 $$1 - 5.42T + 8T^{2}$$
5 $$1 - 16.8T + 125T^{2}$$
13 $$1 + 24.8T + 2.19e3T^{2}$$
17 $$1 - 15.9T + 4.91e3T^{2}$$
19 $$1 + 15.1T + 6.85e3T^{2}$$
23 $$1 - 17.7T + 1.21e4T^{2}$$
29 $$1 + 128.T + 2.43e4T^{2}$$
31 $$1 + 219.T + 2.97e4T^{2}$$
37 $$1 - 92.0T + 5.06e4T^{2}$$
41 $$1 - 459.T + 6.89e4T^{2}$$
43 $$1 - 64.9T + 7.95e4T^{2}$$
47 $$1 + 497.T + 1.03e5T^{2}$$
53 $$1 + 526.T + 1.48e5T^{2}$$
59 $$1 - 578.T + 2.05e5T^{2}$$
61 $$1 - 221.T + 2.26e5T^{2}$$
67 $$1 + 860.T + 3.00e5T^{2}$$
71 $$1 - 580.T + 3.57e5T^{2}$$
73 $$1 + 510.T + 3.89e5T^{2}$$
79 $$1 - 1.03e3T + 4.93e5T^{2}$$
83 $$1 + 606.T + 5.71e5T^{2}$$
89 $$1 - 23.4T + 7.04e5T^{2}$$
97 $$1 + 719.T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$