# Properties

 Degree $2$ Conductor $8034$ Sign $1$ Motivic weight $1$ Primitive yes Self-dual yes Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 2-s − 3-s + 4-s − 2.05·5-s + 6-s + 5.03·7-s − 8-s + 9-s + 2.05·10-s + 3.25·11-s − 12-s + 13-s − 5.03·14-s + 2.05·15-s + 16-s + 0.479·17-s − 18-s + 5.58·19-s − 2.05·20-s − 5.03·21-s − 3.25·22-s − 6.13·23-s + 24-s − 0.793·25-s − 26-s − 27-s + 5.03·28-s + ⋯
 L(s)  = 1 − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.917·5-s + 0.408·6-s + 1.90·7-s − 0.353·8-s + 0.333·9-s + 0.648·10-s + 0.981·11-s − 0.288·12-s + 0.277·13-s − 1.34·14-s + 0.529·15-s + 0.250·16-s + 0.116·17-s − 0.235·18-s + 1.28·19-s − 0.458·20-s − 1.09·21-s − 0.694·22-s − 1.27·23-s + 0.204·24-s − 0.158·25-s − 0.196·26-s − 0.192·27-s + 0.951·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$8034$$    =    $$2 \cdot 3 \cdot 13 \cdot 103$$ Sign: $1$ Motivic weight: $$1$$ Character: $\chi_{8034} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 8034,\ (\ :1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.545843010$$ $$L(\frac12)$$ $$\approx$$ $$1.545843010$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + T$$
3 $$1 + T$$
13 $$1 - T$$
103 $$1 + T$$
good5 $$1 + 2.05T + 5T^{2}$$
7 $$1 - 5.03T + 7T^{2}$$
11 $$1 - 3.25T + 11T^{2}$$
17 $$1 - 0.479T + 17T^{2}$$
19 $$1 - 5.58T + 19T^{2}$$
23 $$1 + 6.13T + 23T^{2}$$
29 $$1 - 3.21T + 29T^{2}$$
31 $$1 - 8.30T + 31T^{2}$$
37 $$1 + 3.88T + 37T^{2}$$
41 $$1 - 5.27T + 41T^{2}$$
43 $$1 - 3.12T + 43T^{2}$$
47 $$1 - 3.55T + 47T^{2}$$
53 $$1 + 6.76T + 53T^{2}$$
59 $$1 - 13.1T + 59T^{2}$$
61 $$1 - 6.84T + 61T^{2}$$
67 $$1 + 3.51T + 67T^{2}$$
71 $$1 - 0.616T + 71T^{2}$$
73 $$1 - 13.2T + 73T^{2}$$
79 $$1 - 2.05T + 79T^{2}$$
83 $$1 + 2.18T + 83T^{2}$$
89 $$1 + 3.35T + 89T^{2}$$
97 $$1 + 1.04T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$