# Properties

 Degree 2 Conductor $2^{5} \cdot 7$ Sign $0.707 + 0.707i$ Motivic weight 2 Primitive yes Self-dual no Analytic rank 0

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## Dirichlet series

 L(s)  = 1 − 2.55i·3-s + 9.86·5-s − 2.64i·7-s + 2.47·9-s + 13.1i·11-s − 5.86·13-s − 25.2i·15-s − 0.570·17-s − 15.6i·19-s − 6.75·21-s + 16.4i·23-s + 72.3·25-s − 29.3i·27-s − 29.7·29-s − 54.8i·31-s + ⋯
 L(s)  = 1 − 0.851i·3-s + 1.97·5-s − 0.377i·7-s + 0.274·9-s + 1.19i·11-s − 0.451·13-s − 1.68i·15-s − 0.0335·17-s − 0.824i·19-s − 0.321·21-s + 0.716i·23-s + 2.89·25-s − 1.08i·27-s − 1.02·29-s − 1.76i·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$224$$    =    $$2^{5} \cdot 7$$ $$\varepsilon$$ = $0.707 + 0.707i$ motivic weight = $$2$$ character : $\chi_{224} (127, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 224,\ (\ :1),\ 0.707 + 0.707i)$$ $$L(\frac{3}{2})$$ $$\approx$$ $$1.97408 - 0.817693i$$ $$L(\frac12)$$ $$\approx$$ $$1.97408 - 0.817693i$$ $$L(2)$$ 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$$
7 $$1 + 2.64iT$$
good3 $$1 + 2.55iT - 9T^{2}$$
5 $$1 - 9.86T + 25T^{2}$$
11 $$1 - 13.1iT - 121T^{2}$$
13 $$1 + 5.86T + 169T^{2}$$
17 $$1 + 0.570T + 289T^{2}$$
19 $$1 + 15.6iT - 361T^{2}$$
23 $$1 - 16.4iT - 529T^{2}$$
29 $$1 + 29.7T + 841T^{2}$$
31 $$1 + 54.8iT - 961T^{2}$$
37 $$1 + 42.0T + 1.36e3T^{2}$$
41 $$1 - 0.773T + 1.68e3T^{2}$$
43 $$1 - 41.7iT - 1.84e3T^{2}$$
47 $$1 - 58.4iT - 2.20e3T^{2}$$
53 $$1 - 5.65T + 2.80e3T^{2}$$
59 $$1 - 42.6iT - 3.48e3T^{2}$$
61 $$1 + 95.9T + 3.72e3T^{2}$$
67 $$1 - 69.8iT - 4.48e3T^{2}$$
71 $$1 + 92.0iT - 5.04e3T^{2}$$
73 $$1 - 9.97T + 5.32e3T^{2}$$
79 $$1 + 20.1iT - 6.24e3T^{2}$$
83 $$1 - 151. iT - 6.88e3T^{2}$$
89 $$1 - 5.79T + 7.92e3T^{2}$$
97 $$1 - 103.T + 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

−12.21126460813623934465187386659, −10.78926818907266438136548721512, −9.739204899573004597039945102407, −9.357300196076271496049307469757, −7.58887242433735179077140025045, −6.83676097967662332829249063265, −5.85835182976470628779714902298, −4.66008621705535079831659915351, −2.40456534071269030521606587998, −1.50098197212991132308958801179, 1.79837948263342960039456953692, 3.29861522692108960745638425725, 5.03800970056913530524778985981, 5.73278127810613671751350318584, 6.80420481128837437157313154216, 8.657437091751355184823086594959, 9.305445802290685809528723453070, 10.28971681132868143795494725762, 10.66995356125795292875394202670, 12.26164882891466705626303154017