# Properties

 Degree 2 Conductor $2^{2} \cdot 3$ Sign $-0.983 - 0.182i$ Motivic weight 15 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (−127. − 128. i)2-s + (−692. + 3.72e3i)3-s + (−7.18 + 3.27e4i)4-s + 2.30e5i·5-s + (5.65e5 − 3.88e5i)6-s + 3.60e6i·7-s + (4.19e6 − 4.19e6i)8-s + (−1.33e7 − 5.15e6i)9-s + (2.95e7 − 2.95e7i)10-s + 5.00e7·11-s + (−1.22e8 − 2.27e7i)12-s + 3.24e7·13-s + (4.61e8 − 4.61e8i)14-s + (−8.58e8 − 1.59e8i)15-s + (−1.07e9 − 4.70e5i)16-s + 2.12e9i·17-s + ⋯
 L(s)  = 1 + (−0.707 − 0.707i)2-s + (−0.182 + 0.983i)3-s + (−0.000219 + 0.999i)4-s + 1.31i·5-s + (0.824 − 0.565i)6-s + 1.65i·7-s + (0.707 − 0.706i)8-s + (−0.933 − 0.359i)9-s + (0.933 − 0.933i)10-s + 0.773·11-s + (−0.983 − 0.182i)12-s + 0.143·13-s + (1.17 − 1.17i)14-s + (−1.29 − 0.241i)15-s + (−0.999 − 0.000438i)16-s + 1.25i·17-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 - 0.182i)\, \overline{\Lambda}(16-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & (-0.983 - 0.182i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$12$$    =    $$2^{2} \cdot 3$$ $$\varepsilon$$ = $-0.983 - 0.182i$ motivic weight = $$15$$ character : $\chi_{12} (11, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 12,\ (\ :15/2),\ -0.983 - 0.182i)$ $L(8)$ $\approx$ $0.0892749 + 0.967691i$ $L(\frac12)$ $\approx$ $0.0892749 + 0.967691i$ $L(\frac{17}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;3\}$, $$F_p$$ is a polynomial of degree 2. If $p \in \{2,\;3\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 $$1 + (127. + 128. i)T$$
3 $$1 + (692. - 3.72e3i)T$$
good5 $$1 - 2.30e5iT - 3.05e10T^{2}$$
7 $$1 - 3.60e6iT - 4.74e12T^{2}$$
11 $$1 - 5.00e7T + 4.17e15T^{2}$$
13 $$1 - 3.24e7T + 5.11e16T^{2}$$
17 $$1 - 2.12e9iT - 2.86e18T^{2}$$
19 $$1 - 2.21e9iT - 1.51e19T^{2}$$
23 $$1 + 1.70e9T + 2.66e20T^{2}$$
29 $$1 + 1.34e11iT - 8.62e21T^{2}$$
31 $$1 - 1.31e11iT - 2.34e22T^{2}$$
37 $$1 - 6.49e11T + 3.33e23T^{2}$$
41 $$1 + 2.08e12iT - 1.55e24T^{2}$$
43 $$1 - 9.33e9iT - 3.17e24T^{2}$$
47 $$1 + 1.60e12T + 1.20e25T^{2}$$
53 $$1 + 1.07e12iT - 7.31e25T^{2}$$
59 $$1 + 1.36e13T + 3.65e26T^{2}$$
61 $$1 - 2.10e13T + 6.02e26T^{2}$$
67 $$1 - 6.54e13iT - 2.46e27T^{2}$$
71 $$1 - 1.13e14T + 5.87e27T^{2}$$
73 $$1 - 7.40e13T + 8.90e27T^{2}$$
79 $$1 + 2.01e13iT - 2.91e28T^{2}$$
83 $$1 + 1.31e14T + 6.11e28T^{2}$$
89 $$1 + 4.75e13iT - 1.74e29T^{2}$$
97 $$1 - 9.60e14T + 6.33e29T^{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

−17.29438180493534177500867741463, −15.66622482152118148930332443218, −14.60578533675557739888328984257, −12.12220803661830598730622875235, −11.06485197349045023574522546716, −9.826223577507927597482567238968, −8.527295054320050824660832530048, −6.15932207475309190987157878443, −3.67350995136903632878156232464, −2.34111584713007542979129055571, 0.55719007597664492381891425793, 1.23200562899961258919165597775, 4.82022510001957056364620180766, 6.70190103754028397639494943327, 7.87518262884372579750497857870, 9.338781455746986373412815530726, 11.27898845172146336289909408205, 13.14479044247297579249106178917, 14.16226366670948186106133012284, 16.46617274163353046710103890980