# Properties

 Degree 2 Conductor $2^{2} \cdot 3 \cdot 23 \cdot 29$ Sign $1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + 3-s − 0.900·5-s + 0.226·7-s + 9-s − 3.23·11-s − 6.29·13-s − 0.900·15-s + 0.102·17-s + 6.74·19-s + 0.226·21-s + 23-s − 4.18·25-s + 27-s − 29-s − 0.708·31-s − 3.23·33-s − 0.203·35-s + 1.94·37-s − 6.29·39-s − 3.13·41-s + 4.38·43-s − 0.900·45-s + 6.47·47-s − 6.94·49-s + 0.102·51-s − 7.91·53-s + 2.91·55-s + ⋯
 L(s)  = 1 + 0.577·3-s − 0.402·5-s + 0.0856·7-s + 0.333·9-s − 0.975·11-s − 1.74·13-s − 0.232·15-s + 0.0249·17-s + 1.54·19-s + 0.0494·21-s + 0.208·23-s − 0.837·25-s + 0.192·27-s − 0.185·29-s − 0.127·31-s − 0.563·33-s − 0.0344·35-s + 0.320·37-s − 1.00·39-s − 0.489·41-s + 0.668·43-s − 0.134·45-s + 0.944·47-s − 0.992·49-s + 0.0143·51-s − 1.08·53-s + 0.392·55-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$8004$$    =    $$2^{2} \cdot 3 \cdot 23 \cdot 29$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{8004} (1, \cdot )$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(2,\ 8004,\ (\ :1/2),\ 1)$ $L(1)$ $\approx$ $1.739951555$ $L(\frac12)$ $\approx$ $1.739951555$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;3,\;23,\;29\}$, $F_p(T) = 1 - a_p T + p T^2 .$If $p \in \{2,\;3,\;23,\;29\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1$$
3 $$1 - T$$
23 $$1 - T$$
29 $$1 + T$$
good5 $$1 + 0.900T + 5T^{2}$$
7 $$1 - 0.226T + 7T^{2}$$
11 $$1 + 3.23T + 11T^{2}$$
13 $$1 + 6.29T + 13T^{2}$$
17 $$1 - 0.102T + 17T^{2}$$
19 $$1 - 6.74T + 19T^{2}$$
31 $$1 + 0.708T + 31T^{2}$$
37 $$1 - 1.94T + 37T^{2}$$
41 $$1 + 3.13T + 41T^{2}$$
43 $$1 - 4.38T + 43T^{2}$$
47 $$1 - 6.47T + 47T^{2}$$
53 $$1 + 7.91T + 53T^{2}$$
59 $$1 - 8.52T + 59T^{2}$$
61 $$1 - 7.66T + 61T^{2}$$
67 $$1 - 12.4T + 67T^{2}$$
71 $$1 - 8.53T + 71T^{2}$$
73 $$1 - 4.34T + 73T^{2}$$
79 $$1 + 7.86T + 79T^{2}$$
83 $$1 - 6.62T + 83T^{2}$$
89 $$1 - 0.363T + 89T^{2}$$
97 $$1 - 9.11T + 97T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}