# Properties

 Degree 2 Conductor $2^{2} \cdot 3 \cdot 7$ Sign $0.895 + 0.444i$ Motivic weight 0 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.5 − 0.866i)3-s + (−0.5 + 0.866i)7-s + (−0.499 + 0.866i)9-s − 13-s + (0.5 − 0.866i)19-s + 0.999·21-s + (−0.5 − 0.866i)25-s + 0.999·27-s + (0.5 + 0.866i)31-s + (0.5 − 0.866i)37-s + (0.5 + 0.866i)39-s − 43-s + (−0.499 − 0.866i)49-s − 0.999·57-s + (−1 + 1.73i)61-s + ⋯
 L(s)  = 1 + (−0.5 − 0.866i)3-s + (−0.5 + 0.866i)7-s + (−0.499 + 0.866i)9-s − 13-s + (0.5 − 0.866i)19-s + 0.999·21-s + (−0.5 − 0.866i)25-s + 0.999·27-s + (0.5 + 0.866i)31-s + (0.5 − 0.866i)37-s + (0.5 + 0.866i)39-s − 43-s + (−0.499 − 0.866i)49-s − 0.999·57-s + (−1 + 1.73i)61-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$84$$    =    $$2^{2} \cdot 3 \cdot 7$$ $$\varepsilon$$ = $0.895 + 0.444i$ motivic weight = $$0$$ character : $\chi_{84} (65, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 84,\ (\ :0),\ 0.895 + 0.444i)$$ $$L(\frac{1}{2})$$ $$\approx$$ $$0.4550711842$$ $$L(\frac12)$$ $$\approx$$ $$0.4550711842$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3,\;7\}$,$$F_p(T)$$ 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 + (0.5 + 0.866i)T$$
7 $$1 + (0.5 - 0.866i)T$$
good5 $$1 + (0.5 + 0.866i)T^{2}$$
11 $$1 + (0.5 - 0.866i)T^{2}$$
13 $$1 + T + T^{2}$$
17 $$1 + (0.5 - 0.866i)T^{2}$$
19 $$1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}$$
23 $$1 + (0.5 + 0.866i)T^{2}$$
29 $$1 - T^{2}$$
31 $$1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}$$
37 $$1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}$$
41 $$1 - T^{2}$$
43 $$1 + T + T^{2}$$
47 $$1 + (0.5 + 0.866i)T^{2}$$
53 $$1 + (0.5 - 0.866i)T^{2}$$
59 $$1 + (0.5 - 0.866i)T^{2}$$
61 $$1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2}$$
67 $$1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}$$
71 $$1 - T^{2}$$
73 $$1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}$$
79 $$1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}$$
83 $$1 - T^{2}$$
89 $$1 + (0.5 + 0.866i)T^{2}$$
97 $$1 - 2T + 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

−14.33197631703146060715751911730, −13.20498120399190008275121892141, −12.27926998190461195602564601842, −11.57081641310592920438165715557, −10.11348932277312344731628512468, −8.796010299104171695587299127572, −7.44725460239841168931737259170, −6.30895696473354994681334896900, −5.07920460323713432335456360487, −2.56497434085969418732679777437, 3.52084052442906496218604031148, 4.89098280278214127535463544377, 6.33268371238830377081639818438, 7.74268924650426005669373898019, 9.514310179830675798001626081592, 10.10977840826758241325303230439, 11.28616606617349332842095808165, 12.33651690769049716182029767608, 13.63570593099300485737171669080, 14.74600246740631530719188745074