# Properties

 Degree 2 Conductor $2^{6} \cdot 3^{2} \cdot 7$ Sign $0.912 - 0.408i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (1.17 − 1.17i)5-s − 7-s + (4.54 + 4.54i)11-s + (2.56 − 2.56i)13-s + 2.05i·17-s + (3.64 + 3.64i)19-s − 2.27i·23-s + 2.21i·25-s + (0.544 + 0.544i)29-s − 10.1i·31-s + (−1.17 + 1.17i)35-s + (4.71 + 4.71i)37-s − 0.487·41-s + (−7.56 + 7.56i)43-s − 0.768·47-s + ⋯
 L(s)  = 1 + (0.527 − 0.527i)5-s − 0.377·7-s + (1.37 + 1.37i)11-s + (0.710 − 0.710i)13-s + 0.497i·17-s + (0.836 + 0.836i)19-s − 0.473i·23-s + 0.443i·25-s + (0.101 + 0.101i)29-s − 1.81i·31-s + (−0.199 + 0.199i)35-s + (0.775 + 0.775i)37-s − 0.0761·41-s + (−1.15 + 1.15i)43-s − 0.112·47-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.912 - 0.408i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.912 - 0.408i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$4032$$    =    $$2^{6} \cdot 3^{2} \cdot 7$$ $$\varepsilon$$ = $0.912 - 0.408i$ motivic weight = $$1$$ character : $\chi_{4032} (3599, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 4032,\ (\ :1/2),\ 0.912 - 0.408i)$ $L(1)$ $\approx$ $2.344005594$ $L(\frac12)$ $\approx$ $2.344005594$ $L(\frac{3}{2})$ 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 + T$$
good5 $$1 + (-1.17 + 1.17i)T - 5iT^{2}$$
11 $$1 + (-4.54 - 4.54i)T + 11iT^{2}$$
13 $$1 + (-2.56 + 2.56i)T - 13iT^{2}$$
17 $$1 - 2.05iT - 17T^{2}$$
19 $$1 + (-3.64 - 3.64i)T + 19iT^{2}$$
23 $$1 + 2.27iT - 23T^{2}$$
29 $$1 + (-0.544 - 0.544i)T + 29iT^{2}$$
31 $$1 + 10.1iT - 31T^{2}$$
37 $$1 + (-4.71 - 4.71i)T + 37iT^{2}$$
41 $$1 + 0.487T + 41T^{2}$$
43 $$1 + (7.56 - 7.56i)T - 43iT^{2}$$
47 $$1 + 0.768T + 47T^{2}$$
53 $$1 + (0.269 - 0.269i)T - 53iT^{2}$$
59 $$1 + (-0.0979 - 0.0979i)T + 59iT^{2}$$
61 $$1 + (7.41 - 7.41i)T - 61iT^{2}$$
67 $$1 + (6.83 + 6.83i)T + 67iT^{2}$$
71 $$1 - 8.66iT - 71T^{2}$$
73 $$1 - 13.6iT - 73T^{2}$$
79 $$1 - 9.29iT - 79T^{2}$$
83 $$1 + (-9.76 + 9.76i)T - 83iT^{2}$$
89 $$1 + 7.27T + 89T^{2}$$
97 $$1 - 10.4T + 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

−8.499067260767964888682606264773, −7.81505738212527514229557604329, −6.96899659538756083319087601597, −6.17609325860565332364729361223, −5.69474307733801068296866884443, −4.63263952922408867963679944259, −3.99199215283738263733368394760, −3.06994851599456193902319706146, −1.80468552097206280763861978958, −1.12522779398748965560093956511, 0.790961129300136761849758157457, 1.84638104359926632899042507062, 3.14660438537510073585305322242, 3.49203845221481255304930644478, 4.61170613270201519842737779352, 5.58543024783913663800070372377, 6.39358771142690650755305628035, 6.65144049522077888680180866416, 7.54011433991089507467182799274, 8.708197884272695246052566482390