# Properties

 Degree 2 Conductor $2 \cdot 3 \cdot 13 \cdot 103$ Sign $-1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 1

# Related objects

## Dirichlet series

 L(s)  = 1 − 2-s + 3-s + 4-s + 1.40·5-s − 6-s − 0.998·7-s − 8-s + 9-s − 1.40·10-s + 4.83·11-s + 12-s + 13-s + 0.998·14-s + 1.40·15-s + 16-s − 2.95·17-s − 18-s + 1.88·19-s + 1.40·20-s − 0.998·21-s − 4.83·22-s − 2.84·23-s − 24-s − 3.01·25-s − 26-s + 27-s − 0.998·28-s + ⋯
 L(s)  = 1 − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.629·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 0.333·9-s − 0.445·10-s + 1.45·11-s + 0.288·12-s + 0.277·13-s + 0.266·14-s + 0.363·15-s + 0.250·16-s − 0.716·17-s − 0.235·18-s + 0.433·19-s + 0.314·20-s − 0.217·21-s − 1.02·22-s − 0.593·23-s − 0.204·24-s − 0.603·25-s − 0.196·26-s + 0.192·27-s − 0.188·28-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

 $$d$$ = $$2$$ $$N$$ = $$8034$$    =    $$2 \cdot 3 \cdot 13 \cdot 103$$ $$\varepsilon$$ = $-1$ motivic weight = $$1$$ character : $\chi_{8034} (1, \cdot )$ primitive : yes self-dual : yes analytic rank = $$1$$ Selberg data = $$(2,\ 8034,\ (\ :1/2),\ -1)$$ $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3,\;13,\;103\}$,$F_p(T) = 1 - a_p T + p T^2 .$If $p \in \{2,\;3,\;13,\;103\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1 + T$$
3 $$1 - T$$
13 $$1 - T$$
103 $$1 + T$$
good5 $$1 - 1.40T + 5T^{2}$$
7 $$1 + 0.998T + 7T^{2}$$
11 $$1 - 4.83T + 11T^{2}$$
17 $$1 + 2.95T + 17T^{2}$$
19 $$1 - 1.88T + 19T^{2}$$
23 $$1 + 2.84T + 23T^{2}$$
29 $$1 + 5.72T + 29T^{2}$$
31 $$1 + 10.1T + 31T^{2}$$
37 $$1 + 4.75T + 37T^{2}$$
41 $$1 + 4.77T + 41T^{2}$$
43 $$1 - 9.75T + 43T^{2}$$
47 $$1 + 2.46T + 47T^{2}$$
53 $$1 + 8.38T + 53T^{2}$$
59 $$1 + 8.26T + 59T^{2}$$
61 $$1 + 6.11T + 61T^{2}$$
67 $$1 + 4.55T + 67T^{2}$$
71 $$1 + 3.66T + 71T^{2}$$
73 $$1 + 13.4T + 73T^{2}$$
79 $$1 + 11.1T + 79T^{2}$$
83 $$1 - 1.76T + 83T^{2}$$
89 $$1 + 0.860T + 89T^{2}$$
97 $$1 + 5.09T + 97T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}