Properties

Degree 2
Conductor $ 2^{2} \cdot 3 $
Sign $-0.232 - 0.972i$
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 + (2.32e7 + 8.29e6i)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 + (−4.15e9 − 1.73e10i)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 + (0.941 + 0.336i)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 + (−0.232 − 0.972i)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

\[\begin{aligned} \Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.232 - 0.972i)\, \overline{\Lambda}(20-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & (-0.232 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(12\)    =    \(2^{2} \cdot 3\)
\( \varepsilon \)  =  $-0.232 - 0.972i$
motivic weight  =  \(19\)
character  :  $\chi_{12} (11, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 12,\ (\ :19/2),\ -0.232 - 0.972i)$
$L(10)$  $\approx$  $0.5151243510$
$L(\frac12)$  $\approx$  $0.5151243510$
$L(\frac{21}{2})$   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

−16.35219371358912854573135941474, −14.46499919491340902358926020224, −12.45236472094745806071237051724, −11.45937459266538358285103151921, −10.19618688943886888057380028176, −9.050476635341304302257136520060, −6.93465328593272969046210493916, −4.80072248003983213466976828912, −3.33081347854913436143987992782, −1.25080057586313120908499009284, 0.25629224598614410909920647828, 1.60441105577807598169596261292, 4.74982046692522468200896381738, 6.20943451020240394180697708529, 7.32033055986811253671029871104, 8.954342401244154937914102156405, 10.62259386551977676966666248906, 12.20685265991111872881744603688, 13.80862685257145522667867118942, 15.29085782365172617351669580388

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