# Properties

 Degree 2 Conductor $2^{5} \cdot 7$ Sign $-0.588 - 0.808i$ Motivic weight 2 Primitive yes Self-dual no Analytic rank 0

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## Dirichlet series

 L(s)  = 1 + (−1.03 + 1.71i)2-s + (0.406 + 0.982i)3-s + (−1.87 − 3.53i)4-s + (0.691 − 1.66i)5-s + (−2.10 − 0.315i)6-s + (−1.87 + 1.87i)7-s + (7.98 + 0.433i)8-s + (5.56 − 5.56i)9-s + (2.14 + 2.90i)10-s + (−6.83 + 16.4i)11-s + (2.70 − 3.27i)12-s + (7.71 + 18.6i)13-s + (−1.27 − 5.13i)14-s + 1.92·15-s + (−8.97 + 13.2i)16-s + 1.25i·17-s + ⋯
 L(s)  = 1 + (−0.515 + 0.856i)2-s + (0.135 + 0.327i)3-s + (−0.468 − 0.883i)4-s + (0.138 − 0.333i)5-s + (−0.350 − 0.0525i)6-s + (−0.267 + 0.267i)7-s + (0.998 + 0.0541i)8-s + (0.618 − 0.618i)9-s + (0.214 + 0.290i)10-s + (−0.621 + 1.49i)11-s + (0.225 − 0.273i)12-s + (0.593 + 1.43i)13-s + (−0.0912 − 0.366i)14-s + 0.128·15-s + (−0.561 + 0.827i)16-s + 0.0737i·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.588 - 0.808i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.588 - 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$224$$    =    $$2^{5} \cdot 7$$ $$\varepsilon$$ = $-0.588 - 0.808i$ motivic weight = $$2$$ character : $\chi_{224} (43, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 224,\ (\ :1),\ -0.588 - 0.808i)$$ $$L(\frac{3}{2})$$ $$\approx$$ $$0.489703 + 0.962671i$$ $$L(\frac12)$$ $$\approx$$ $$0.489703 + 0.962671i$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;7\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{2,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1 + (1.03 - 1.71i)T$$
7 $$1 + (1.87 - 1.87i)T$$
good3 $$1 + (-0.406 - 0.982i)T + (-6.36 + 6.36i)T^{2}$$
5 $$1 + (-0.691 + 1.66i)T + (-17.6 - 17.6i)T^{2}$$
11 $$1 + (6.83 - 16.4i)T + (-85.5 - 85.5i)T^{2}$$
13 $$1 + (-7.71 - 18.6i)T + (-119. + 119. i)T^{2}$$
17 $$1 - 1.25iT - 289T^{2}$$
19 $$1 + (8.00 - 3.31i)T + (255. - 255. i)T^{2}$$
23 $$1 + (-4.46 - 4.46i)T + 529iT^{2}$$
29 $$1 + (15.0 - 6.21i)T + (594. - 594. i)T^{2}$$
31 $$1 - 1.05iT - 961T^{2}$$
37 $$1 + (-17.3 + 41.7i)T + (-968. - 968. i)T^{2}$$
41 $$1 + (41.0 - 41.0i)T - 1.68e3iT^{2}$$
43 $$1 + (18.8 - 45.3i)T + (-1.30e3 - 1.30e3i)T^{2}$$
47 $$1 + 16.4T + 2.20e3T^{2}$$
53 $$1 + (-2.31 - 0.957i)T + (1.98e3 + 1.98e3i)T^{2}$$
59 $$1 + (-60.1 - 24.9i)T + (2.46e3 + 2.46e3i)T^{2}$$
61 $$1 + (-111. + 46.3i)T + (2.63e3 - 2.63e3i)T^{2}$$
67 $$1 + (-12.6 - 30.5i)T + (-3.17e3 + 3.17e3i)T^{2}$$
71 $$1 + (-19.6 + 19.6i)T - 5.04e3iT^{2}$$
73 $$1 + (-60.5 + 60.5i)T - 5.32e3iT^{2}$$
79 $$1 + 46.4T + 6.24e3T^{2}$$
83 $$1 + (12.5 - 5.21i)T + (4.87e3 - 4.87e3i)T^{2}$$
89 $$1 + (31.5 + 31.5i)T + 7.92e3iT^{2}$$
97 $$1 + 36.2T + 9.40e3T^{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

−12.59996308959966858433294495713, −11.20192204418527485788002186568, −9.915854456695570677938594166963, −9.461357386615264949518539763943, −8.552243967664778833148099351150, −7.21776230444938831161314851829, −6.50203116297918621539619101324, −5.08022145022648682823818625074, −4.11068954034454996498900532902, −1.69693721631185057212786719608, 0.73090591788205806345096474052, 2.56173350545315549108517299495, 3.61877945320571728235022685529, 5.29825900469378695737445618581, 6.81925781261487767156917237525, 8.058645009948864775601944817038, 8.558952056627369700456327168957, 10.17259360628604194872443237886, 10.55514148766404615361860555357, 11.44412931896687883993126675651