# Properties

 Degree 2 Conductor $2^{4} \cdot 3^{2} \cdot 7$ Sign $-0.991 - 0.126i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

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## Dirichlet series

 L(s)  = 1 + (1.5 + 0.866i)5-s − 2.64·7-s + (−3.96 + 2.29i)11-s − 3.46i·13-s + (−4.5 + 2.59i)17-s + (−1.32 + 2.29i)19-s + (−3.96 − 2.29i)23-s + (−1 − 1.73i)25-s + (1.32 + 2.29i)31-s + (−3.96 − 2.29i)35-s + (−3.5 + 6.06i)37-s − 3.46i·41-s − 9.16i·43-s + (−3.96 + 6.87i)47-s + 7.00·49-s + ⋯
 L(s)  = 1 + (0.670 + 0.387i)5-s − 0.999·7-s + (−1.19 + 0.690i)11-s − 0.960i·13-s + (−1.09 + 0.630i)17-s + (−0.303 + 0.525i)19-s + (−0.827 − 0.477i)23-s + (−0.200 − 0.346i)25-s + (0.237 + 0.411i)31-s + (−0.670 − 0.387i)35-s + (−0.575 + 0.996i)37-s − 0.541i·41-s − 1.39i·43-s + (−0.578 + 1.00i)47-s + 49-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$1008$$    =    $$2^{4} \cdot 3^{2} \cdot 7$$ $$\varepsilon$$ = $-0.991 - 0.126i$ motivic weight = $$1$$ character : $\chi_{1008} (271, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 1008,\ (\ :1/2),\ -0.991 - 0.126i)$$ $$L(1)$$ $$\approx$$ $$0.2476878935$$ $$L(\frac12)$$ $$\approx$$ $$0.2476878935$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3,\;7\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1$$
3 $$1$$
7 $$1 + 2.64T$$
good5 $$1 + (-1.5 - 0.866i)T + (2.5 + 4.33i)T^{2}$$
11 $$1 + (3.96 - 2.29i)T + (5.5 - 9.52i)T^{2}$$
13 $$1 + 3.46iT - 13T^{2}$$
17 $$1 + (4.5 - 2.59i)T + (8.5 - 14.7i)T^{2}$$
19 $$1 + (1.32 - 2.29i)T + (-9.5 - 16.4i)T^{2}$$
23 $$1 + (3.96 + 2.29i)T + (11.5 + 19.9i)T^{2}$$
29 $$1 + 29T^{2}$$
31 $$1 + (-1.32 - 2.29i)T + (-15.5 + 26.8i)T^{2}$$
37 $$1 + (3.5 - 6.06i)T + (-18.5 - 32.0i)T^{2}$$
41 $$1 + 3.46iT - 41T^{2}$$
43 $$1 + 9.16iT - 43T^{2}$$
47 $$1 + (3.96 - 6.87i)T + (-23.5 - 40.7i)T^{2}$$
53 $$1 + (-1.5 - 2.59i)T + (-26.5 + 45.8i)T^{2}$$
59 $$1 + (3.96 + 6.87i)T + (-29.5 + 51.0i)T^{2}$$
61 $$1 + (-1.5 - 0.866i)T + (30.5 + 52.8i)T^{2}$$
67 $$1 + (3.96 - 2.29i)T + (33.5 - 58.0i)T^{2}$$
71 $$1 - 9.16iT - 71T^{2}$$
73 $$1 + (4.5 - 2.59i)T + (36.5 - 63.2i)T^{2}$$
79 $$1 + (-3.96 - 2.29i)T + (39.5 + 68.4i)T^{2}$$
83 $$1 + 83T^{2}$$
89 $$1 + (1.5 + 0.866i)T + (44.5 + 77.0i)T^{2}$$
97 $$1 - 3.46iT - 97T^{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

−10.28279798689886151661312613252, −9.842320007172630554727307762236, −8.682047818553468861982044978363, −7.901360579886362904830349292478, −6.84304158859718711240011749337, −6.15918770743544933030043835052, −5.33877641167049481110865710861, −4.13955444668826971207997872736, −2.92123545981488831272916035124, −2.09701923026291125310806339808, 0.10050117155282918327110425986, 2.02112644427958919218154372903, 3.01945934665557814152385975183, 4.27666683240043686262635125860, 5.31554391104544217797693165917, 6.13501858343338108528619396526, 6.89873427578238172091999417391, 7.938168923503822329196980092494, 9.012940866819939865859100205414, 9.423801890902206756870536684151