# Properties

 Degree 2 Conductor $2^{2} \cdot 3 \cdot 167$ Sign $0.841 - 0.540i$ Motivic weight 0 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.415 − 0.909i)2-s + (−0.142 + 0.989i)3-s + (−0.654 − 0.755i)4-s + (0.841 + 0.540i)6-s + 1.08i·7-s + (−0.959 + 0.281i)8-s + (−0.959 − 0.281i)9-s + 1.30·11-s + (0.841 − 0.540i)12-s + (0.983 + 0.449i)14-s + (−0.142 + 0.989i)16-s + (−0.654 + 0.755i)18-s + 0.563i·19-s + (−1.07 − 0.153i)21-s + (0.544 − 1.19i)22-s + ⋯
 L(s)  = 1 + (0.415 − 0.909i)2-s + (−0.142 + 0.989i)3-s + (−0.654 − 0.755i)4-s + (0.841 + 0.540i)6-s + 1.08i·7-s + (−0.959 + 0.281i)8-s + (−0.959 − 0.281i)9-s + 1.30·11-s + (0.841 − 0.540i)12-s + (0.983 + 0.449i)14-s + (−0.142 + 0.989i)16-s + (−0.654 + 0.755i)18-s + 0.563i·19-s + (−1.07 − 0.153i)21-s + (0.544 − 1.19i)22-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.841 - 0.540i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.841 - 0.540i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$2004$$    =    $$2^{2} \cdot 3 \cdot 167$$ $$\varepsilon$$ = $0.841 - 0.540i$ motivic weight = $$0$$ character : $\chi_{2004} (2003, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 2004,\ (\ :0),\ 0.841 - 0.540i)$$ $$L(\frac{1}{2})$$ $$\approx$$ $$1.190431511$$ $$L(\frac12)$$ $$\approx$$ $$1.190431511$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3,\;167\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{2,\;3,\;167\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1 + (-0.415 + 0.909i)T$$
3 $$1 + (0.142 - 0.989i)T$$
167 $$1 - T$$
good5 $$1 + T^{2}$$
7 $$1 - 1.08iT - T^{2}$$
11 $$1 - 1.30T + T^{2}$$
13 $$1 - T^{2}$$
17 $$1 + T^{2}$$
19 $$1 - 0.563iT - T^{2}$$
23 $$1 - T^{2}$$
29 $$1 - 1.08iT - T^{2}$$
31 $$1 - 1.81iT - T^{2}$$
37 $$1 - T^{2}$$
41 $$1 + T^{2}$$
43 $$1 + T^{2}$$
47 $$1 - 1.91T + T^{2}$$
53 $$1 + T^{2}$$
59 $$1 - T^{2}$$
61 $$1 - 0.284T + T^{2}$$
67 $$1 + T^{2}$$
71 $$1 - T^{2}$$
73 $$1 - T^{2}$$
79 $$1 + T^{2}$$
83 $$1 - T^{2}$$
89 $$1 + 1.97iT - T^{2}$$
97 $$1 + 1.30T + T^{2}$$
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\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}

## Imaginary part of the first few zeros on the critical line

−9.373198838361257665282238493173, −9.013196219139209236541804679348, −8.361252873877966315066126284997, −6.76464643771582641455260546907, −5.82707997005026235943988095459, −5.35484805609351314044853233179, −4.35009334602545203064514680570, −3.64255122552273147148191816058, −2.79893431239952689024205794906, −1.58219327958329133255244849674, 0.821356546807012871468183666687, 2.39591821652881302389623929964, 3.80668576115319999058876370672, 4.29228105616542668826509004898, 5.58508673427531047986895518467, 6.25130226029610824114877687755, 6.92353123945210441463524168047, 7.54108521378009775109095094792, 8.132576618358243388597939307112, 9.099945123526947744198126803519