Properties

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

Origins

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Normalization:  

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

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

Graph of the $Z$-function along the critical line