# 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 − 2.61·5-s + 6-s − 0.977·7-s − 8-s + 9-s + 2.61·10-s − 0.240·11-s − 12-s + 13-s + 0.977·14-s + 2.61·15-s + 16-s − 4.91·17-s − 18-s − 1.97·19-s − 2.61·20-s + 0.977·21-s + 0.240·22-s − 0.244·23-s + 24-s + 1.84·25-s − 26-s − 27-s − 0.977·28-s + ⋯
 L(s)  = 1 − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.17·5-s + 0.408·6-s − 0.369·7-s − 0.353·8-s + 0.333·9-s + 0.827·10-s − 0.0725·11-s − 0.288·12-s + 0.277·13-s + 0.261·14-s + 0.675·15-s + 0.250·16-s − 1.19·17-s − 0.235·18-s − 0.453·19-s − 0.585·20-s + 0.213·21-s + 0.0513·22-s − 0.0510·23-s + 0.204·24-s + 0.369·25-s − 0.196·26-s − 0.192·27-s − 0.184·28-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 $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 $$1 + T$$
3 $$1 + T$$
13 $$1 - T$$
103 $$1 - T$$
good5 $$1 + 2.61T + 5T^{2}$$
7 $$1 + 0.977T + 7T^{2}$$
11 $$1 + 0.240T + 11T^{2}$$
17 $$1 + 4.91T + 17T^{2}$$
19 $$1 + 1.97T + 19T^{2}$$
23 $$1 + 0.244T + 23T^{2}$$
29 $$1 - 8.09T + 29T^{2}$$
31 $$1 + 7.15T + 31T^{2}$$
37 $$1 - 5.41T + 37T^{2}$$
41 $$1 - 10.7T + 41T^{2}$$
43 $$1 + 5.03T + 43T^{2}$$
47 $$1 - 9.26T + 47T^{2}$$
53 $$1 - 9.79T + 53T^{2}$$
59 $$1 + 7.05T + 59T^{2}$$
61 $$1 - 3.55T + 61T^{2}$$
67 $$1 + 14.2T + 67T^{2}$$
71 $$1 - 9.42T + 71T^{2}$$
73 $$1 - 0.976T + 73T^{2}$$
79 $$1 + 8.87T + 79T^{2}$$
83 $$1 - 13.3T + 83T^{2}$$
89 $$1 + 0.490T + 89T^{2}$$
97 $$1 - 14.7T + 97T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}