Properties

Degree 2
Conductor $ 2^{2} \cdot 3 $
Sign $-0.710 + 0.703i$
Motivic weight 19
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−627. − 362. i)2-s + (8.68e3 + 3.29e4i)3-s + (2.61e5 + 4.54e5i)4-s − 6.03e6i·5-s + (6.49e6 − 2.38e7i)6-s + 9.28e7i·7-s + (2.07e5 − 3.79e8i)8-s + (−1.01e9 + 5.72e8i)9-s + (−2.18e9 + 3.78e9i)10-s − 6.13e9·11-s + (−1.26e10 + 1.25e10i)12-s + 4.99e10·13-s + (3.36e10 − 5.82e10i)14-s + (1.98e11 − 5.23e10i)15-s + (−1.37e11 + 2.37e11i)16-s + 2.01e11i·17-s + ⋯
L(s)  = 1  + (−0.865 − 0.500i)2-s + (0.254 + 0.967i)3-s + (0.499 + 0.866i)4-s − 1.38i·5-s + (0.263 − 0.964i)6-s + 0.870i·7-s + (0.000545 − 0.999i)8-s + (−0.870 + 0.492i)9-s + (−0.690 + 1.19i)10-s − 0.784·11-s + (−0.710 + 0.703i)12-s + 1.30·13-s + (0.435 − 0.753i)14-s + (1.33 − 0.351i)15-s + (−0.500 + 0.865i)16-s + 0.412i·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.710 + 0.703i)\, \overline{\Lambda}(20-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & (-0.710 + 0.703i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(12\)    =    \(2^{2} \cdot 3\)
\( \varepsilon \)  =  $-0.710 + 0.703i$
motivic weight  =  \(19\)
character  :  $\chi_{12} (11, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 12,\ (\ :19/2),\ -0.710 + 0.703i)$
$L(10)$  $\approx$  $0.4408918534$
$L(\frac12)$  $\approx$  $0.4408918534$
$L(\frac{21}{2})$   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 \( 1 + (627. + 362. i)T \)
3 \( 1 + (-8.68e3 - 3.29e4i)T \)
good5 \( 1 + 6.03e6iT - 1.90e13T^{2} \)
7 \( 1 - 9.28e7iT - 1.13e16T^{2} \)
11 \( 1 + 6.13e9T + 6.11e19T^{2} \)
13 \( 1 - 4.99e10T + 1.46e21T^{2} \)
17 \( 1 - 2.01e11iT - 2.39e23T^{2} \)
19 \( 1 + 4.67e11iT - 1.97e24T^{2} \)
23 \( 1 + 8.46e12T + 7.46e25T^{2} \)
29 \( 1 - 2.39e13iT - 6.10e27T^{2} \)
31 \( 1 + 1.41e14iT - 2.16e28T^{2} \)
37 \( 1 + 1.37e15T + 6.24e29T^{2} \)
41 \( 1 + 3.22e15iT - 4.39e30T^{2} \)
43 \( 1 + 5.84e15iT - 1.08e31T^{2} \)
47 \( 1 + 1.11e16T + 5.88e31T^{2} \)
53 \( 1 + 3.17e16iT - 5.77e32T^{2} \)
59 \( 1 - 1.59e16T + 4.42e33T^{2} \)
61 \( 1 - 1.40e17T + 8.34e33T^{2} \)
67 \( 1 - 3.36e16iT - 4.95e34T^{2} \)
71 \( 1 + 6.45e17T + 1.49e35T^{2} \)
73 \( 1 + 6.40e17T + 2.53e35T^{2} \)
79 \( 1 + 3.85e17iT - 1.13e36T^{2} \)
83 \( 1 - 5.19e17T + 2.90e36T^{2} \)
89 \( 1 - 5.87e17iT - 1.09e37T^{2} \)
97 \( 1 + 4.81e18T + 5.60e37T^{2} \)
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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

−15.63135350527899007358146246252, −13.22186298208320411918120693699, −11.84115708533968029572619998777, −10.35209128250543943252318629240, −8.904249734862061996209587742207, −8.381912173768112592054986078548, −5.46702080236440612353464798824, −3.76684058797179403189858618887, −1.99871373631969360369405867854, −0.18298350708844538945477891312, 1.41837751060897160255949145847, 3.01236663856622698184619463465, 6.15626751508506223477278760567, 7.14797888397952129366237018358, 8.216365374303622088916151231846, 10.24061898430553254699949760550, 11.30770809203492987057578552821, 13.60249468522568144850710384178, 14.54628643433639213643352117754, 16.06506240917298621766821681217

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