Properties

Degree 2
Conductor $ 2^{2} \cdot 3 $
Sign $0.724 - 0.689i$
Motivic weight 15
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (166. + 71.1i)2-s + (8.61 − 3.78e3i)3-s + (2.26e4 + 2.36e4i)4-s + 9.97e4i·5-s + (2.71e5 − 6.29e5i)6-s + 1.78e6i·7-s + (2.08e6 + 5.55e6i)8-s + (−1.43e7 − 6.52e4i)9-s + (−7.10e6 + 1.66e7i)10-s + 1.16e8·11-s + (8.99e7 − 8.55e7i)12-s + 2.21e8·13-s + (−1.26e8 + 2.96e8i)14-s + (3.77e8 + 8.59e5i)15-s + (−4.88e7 + 1.07e9i)16-s − 1.04e9i·17-s + ⋯
L(s)  = 1  + (0.919 + 0.393i)2-s + (0.00227 − 0.999i)3-s + (0.690 + 0.723i)4-s + 0.571i·5-s + (0.395 − 0.918i)6-s + 0.818i·7-s + (0.350 + 0.936i)8-s + (−0.999 − 0.00454i)9-s + (−0.224 + 0.525i)10-s + 1.79·11-s + (0.724 − 0.689i)12-s + 0.978·13-s + (−0.321 + 0.752i)14-s + (0.571 + 0.00129i)15-s + (−0.0455 + 0.998i)16-s − 0.619i·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.724 - 0.689i)\, \overline{\Lambda}(16-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & (0.724 - 0.689i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(12\)    =    \(2^{2} \cdot 3\)
\( \varepsilon \)  =  $0.724 - 0.689i$
motivic weight  =  \(15\)
character  :  $\chi_{12} (11, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 12,\ (\ :15/2),\ 0.724 - 0.689i)$
$L(8)$  $\approx$  $3.22377 + 1.28827i$
$L(\frac12)$  $\approx$  $3.22377 + 1.28827i$
$L(\frac{17}{2})$   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 \( 1 + (-166. - 71.1i)T \)
3 \( 1 + (-8.61 + 3.78e3i)T \)
good5 \( 1 - 9.97e4iT - 3.05e10T^{2} \)
7 \( 1 - 1.78e6iT - 4.74e12T^{2} \)
11 \( 1 - 1.16e8T + 4.17e15T^{2} \)
13 \( 1 - 2.21e8T + 5.11e16T^{2} \)
17 \( 1 + 1.04e9iT - 2.86e18T^{2} \)
19 \( 1 - 4.40e9iT - 1.51e19T^{2} \)
23 \( 1 + 1.18e10T + 2.66e20T^{2} \)
29 \( 1 - 3.11e10iT - 8.62e21T^{2} \)
31 \( 1 + 8.47e10iT - 2.34e22T^{2} \)
37 \( 1 + 5.63e11T + 3.33e23T^{2} \)
41 \( 1 + 1.67e12iT - 1.55e24T^{2} \)
43 \( 1 - 6.70e11iT - 3.17e24T^{2} \)
47 \( 1 + 4.09e12T + 1.20e25T^{2} \)
53 \( 1 + 1.54e13iT - 7.31e25T^{2} \)
59 \( 1 - 1.68e13T + 3.65e26T^{2} \)
61 \( 1 + 2.24e13T + 6.02e26T^{2} \)
67 \( 1 + 8.16e11iT - 2.46e27T^{2} \)
71 \( 1 + 2.78e13T + 5.87e27T^{2} \)
73 \( 1 + 9.23e13T + 8.90e27T^{2} \)
79 \( 1 - 1.17e14iT - 2.91e28T^{2} \)
83 \( 1 - 3.73e14T + 6.11e28T^{2} \)
89 \( 1 - 9.06e13iT - 1.74e29T^{2} \)
97 \( 1 - 2.89e14T + 6.33e29T^{2} \)
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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

−16.44524344755351873017215139091, −14.74203826705785515826344565628, −13.90590774916411669213959718393, −12.32884876318311318373464140367, −11.45436217708099021700931790609, −8.575039157838278774941458430478, −6.86586087414299616347533277134, −5.88974867898183727315054388981, −3.47776663154997411412016478763, −1.78525848636299174880348817249, 1.14869408976486925641633082849, 3.59994970956373672915363763127, 4.56077640050637668241960708229, 6.35711492420843131912434348045, 9.048530987149966142075444789874, 10.59364821727321366802490077746, 11.79728883140808715850652904420, 13.55158951079959765000228062651, 14.67521391093183829232818447484, 16.06283568153930093951077698958

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