Properties

Degree 2
Conductor $ 2^{2} \cdot 3 $
Sign $-0.875 - 0.483i$
Motivic weight 19
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (395. − 606. i)2-s + (2.71e4 − 2.06e4i)3-s + (−2.11e5 − 4.79e5i)4-s + 7.30e5i·5-s + (−1.79e6 − 2.46e7i)6-s + 7.02e6i·7-s + (−3.74e8 − 6.12e7i)8-s + (3.09e8 − 1.12e9i)9-s + (4.43e8 + 2.88e8i)10-s − 9.08e9·11-s + (−1.56e10 − 8.64e9i)12-s − 2.88e10·13-s + (4.26e9 + 2.77e9i)14-s + (1.50e10 + 1.98e10i)15-s + (−1.85e11 + 2.03e11i)16-s − 3.88e11i·17-s + ⋯
L(s)  = 1  + (0.546 − 0.837i)2-s + (0.795 − 0.605i)3-s + (−0.403 − 0.914i)4-s + 0.167i·5-s + (−0.0727 − 0.997i)6-s + 0.0658i·7-s + (−0.986 − 0.161i)8-s + (0.266 − 0.963i)9-s + (0.140 + 0.0913i)10-s − 1.16·11-s + (−0.875 − 0.483i)12-s − 0.755·13-s + (0.0551 + 0.0359i)14-s + (0.101 + 0.133i)15-s + (−0.673 + 0.738i)16-s − 0.794i·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.875 - 0.483i)\, \overline{\Lambda}(20-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & (-0.875 - 0.483i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(12\)    =    \(2^{2} \cdot 3\)
\( \varepsilon \)  =  $-0.875 - 0.483i$
motivic weight  =  \(19\)
character  :  $\chi_{12} (11, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 12,\ (\ :19/2),\ -0.875 - 0.483i)$
$L(10)$  $\approx$  $1.906594585$
$L(\frac12)$  $\approx$  $1.906594585$
$L(\frac{21}{2})$   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 \( 1 + (-395. + 606. i)T \)
3 \( 1 + (-2.71e4 + 2.06e4i)T \)
good5 \( 1 - 7.30e5iT - 1.90e13T^{2} \)
7 \( 1 - 7.02e6iT - 1.13e16T^{2} \)
11 \( 1 + 9.08e9T + 6.11e19T^{2} \)
13 \( 1 + 2.88e10T + 1.46e21T^{2} \)
17 \( 1 + 3.88e11iT - 2.39e23T^{2} \)
19 \( 1 + 1.03e12iT - 1.97e24T^{2} \)
23 \( 1 + 2.94e12T + 7.46e25T^{2} \)
29 \( 1 - 8.24e13iT - 6.10e27T^{2} \)
31 \( 1 - 2.06e12iT - 2.16e28T^{2} \)
37 \( 1 + 1.42e15T + 6.24e29T^{2} \)
41 \( 1 - 1.17e15iT - 4.39e30T^{2} \)
43 \( 1 + 3.15e15iT - 1.08e31T^{2} \)
47 \( 1 - 8.86e15T + 5.88e31T^{2} \)
53 \( 1 + 3.81e16iT - 5.77e32T^{2} \)
59 \( 1 - 1.21e17T + 4.42e33T^{2} \)
61 \( 1 + 1.00e17T + 8.34e33T^{2} \)
67 \( 1 + 3.94e17iT - 4.95e34T^{2} \)
71 \( 1 - 5.81e16T + 1.49e35T^{2} \)
73 \( 1 + 3.90e17T + 2.53e35T^{2} \)
79 \( 1 + 1.66e18iT - 1.13e36T^{2} \)
83 \( 1 + 1.76e17T + 2.90e36T^{2} \)
89 \( 1 + 1.08e18iT - 1.09e37T^{2} \)
97 \( 1 - 8.25e18T + 5.60e37T^{2} \)
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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

−14.29538245363366731856835031878, −13.20689572908797958007239341629, −12.09921096635974458045894752691, −10.39583346560581455156549713498, −8.911863511490635140584800983057, −7.08765446476829443514803621355, −5.06984180923199227730255334203, −3.16015180993997891332040943476, −2.15531806938441388380511728809, −0.43210643967089306883343196578, 2.59097109089216957349322782246, 4.10180278931081193483905166785, 5.43169326963752723005953856000, 7.50409847069905234054811490658, 8.621962282845163369889032211404, 10.25955776770127045516468367855, 12.53583744451960645542446271032, 13.78713060941788344393580790522, 14.95714856140412615141429835398, 15.92869890908711244421711403460

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