Properties

Degree 2
Conductor $ 2^{2} \cdot 3 $
Sign $0.882 + 0.470i$
Motivic weight 19
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (246. + 680. i)2-s + (−1.28e4 − 3.15e4i)3-s + (−4.02e5 + 3.35e5i)4-s − 2.25e5i·5-s + (1.83e7 − 1.65e7i)6-s + 9.84e7i·7-s + (−3.27e8 − 1.91e8i)8-s + (−8.32e8 + 8.11e8i)9-s + (1.53e8 − 5.55e7i)10-s − 6.04e9·11-s + (1.57e10 + 8.41e9i)12-s + 4.74e10·13-s + (−6.70e10 + 2.42e10i)14-s + (−7.12e9 + 2.89e9i)15-s + (4.98e10 − 2.70e11i)16-s − 8.55e11i·17-s + ⋯
L(s)  = 1  + (0.340 + 0.940i)2-s + (−0.376 − 0.926i)3-s + (−0.768 + 0.639i)4-s − 0.0516i·5-s + (0.742 − 0.669i)6-s + 0.922i·7-s + (−0.863 − 0.505i)8-s + (−0.716 + 0.698i)9-s + (0.0485 − 0.0175i)10-s − 0.773·11-s + (0.882 + 0.470i)12-s + 1.24·13-s + (−0.867 + 0.313i)14-s + (−0.0478 + 0.0194i)15-s + (0.181 − 0.983i)16-s − 1.74i·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.882 + 0.470i)\, \overline{\Lambda}(20-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & (0.882 + 0.470i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(12\)    =    \(2^{2} \cdot 3\)
\( \varepsilon \)  =  $0.882 + 0.470i$
motivic weight  =  \(19\)
character  :  $\chi_{12} (11, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 12,\ (\ :19/2),\ 0.882 + 0.470i)$
$L(10)$  $\approx$  $1.368691195$
$L(\frac12)$  $\approx$  $1.368691195$
$L(\frac{21}{2})$   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 \( 1 + (-246. - 680. i)T \)
3 \( 1 + (1.28e4 + 3.15e4i)T \)
good5 \( 1 + 2.25e5iT - 1.90e13T^{2} \)
7 \( 1 - 9.84e7iT - 1.13e16T^{2} \)
11 \( 1 + 6.04e9T + 6.11e19T^{2} \)
13 \( 1 - 4.74e10T + 1.46e21T^{2} \)
17 \( 1 + 8.55e11iT - 2.39e23T^{2} \)
19 \( 1 + 3.94e11iT - 1.97e24T^{2} \)
23 \( 1 - 5.41e10T + 7.46e25T^{2} \)
29 \( 1 - 3.42e12iT - 6.10e27T^{2} \)
31 \( 1 + 2.38e14iT - 2.16e28T^{2} \)
37 \( 1 - 4.65e14T + 6.24e29T^{2} \)
41 \( 1 - 2.62e14iT - 4.39e30T^{2} \)
43 \( 1 - 2.87e13iT - 1.08e31T^{2} \)
47 \( 1 - 5.76e15T + 5.88e31T^{2} \)
53 \( 1 + 3.52e16iT - 5.77e32T^{2} \)
59 \( 1 + 1.05e17T + 4.42e33T^{2} \)
61 \( 1 + 5.09e16T + 8.34e33T^{2} \)
67 \( 1 + 1.29e17iT - 4.95e34T^{2} \)
71 \( 1 + 4.53e17T + 1.49e35T^{2} \)
73 \( 1 - 8.94e17T + 2.53e35T^{2} \)
79 \( 1 + 6.91e17iT - 1.13e36T^{2} \)
83 \( 1 + 2.08e18T + 2.90e36T^{2} \)
89 \( 1 - 2.97e18iT - 1.09e37T^{2} \)
97 \( 1 - 9.21e17T + 5.60e37T^{2} \)
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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

−15.47404750235911325862130116064, −13.82460348563722354939700832289, −12.84450579309950299648829822384, −11.49713371138124452766563609600, −8.943169097957834328335225683664, −7.64372711738779709065082467912, −6.22082373183003506922940704785, −5.10067196155315072829721580689, −2.72590893557139665523002948449, −0.49755183416833571880044784868, 1.13197910338428913054163704034, 3.31729948392858858458818783252, 4.39673674563355270718124842130, 5.95035215416623273799055469925, 8.659622889780519162898914529759, 10.42591189481778777477336962421, 10.84330756431619659805977063425, 12.63909628709417061928890371232, 14.04254192581130957289641377305, 15.43381824531141776997628469201

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