Properties

Degree 2
Conductor $ 2^{2} \cdot 3 $
Sign $0.302 - 0.952i$
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 + (−2.46e7 + 9.68e5i)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 + (5.41e9 − 1.70e10i)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 + (−0.999 + 0.0392i)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 + (0.302 − 0.952i)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

\[\begin{aligned} \Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.302 - 0.952i)\, \overline{\Lambda}(20-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & (0.302 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(12\)    =    \(2^{2} \cdot 3\)
\( \varepsilon \)  =  $0.302 - 0.952i$
motivic weight  =  \(19\)
character  :  $\chi_{12} (11, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 12,\ (\ :19/2),\ 0.302 - 0.952i)$
$L(10)$  $\approx$  $1.844287021$
$L(\frac12)$  $\approx$  $1.844287021$
$L(\frac{21}{2})$   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

−15.99077889405484318075751336015, −14.47266092995944937340509424718, −13.64202048722460671818185589014, −10.96946581297834792666533487313, −9.638169773785422327463671924568, −8.467572633518650300208319971682, −6.83227617511172053752460626922, −4.99919222609535503262279963651, −3.66931746599993416237722226823, −0.893463380071435957067856761693, 1.02895650902107728308762832913, 2.19394943931718253813683280394, 3.68986762403361364378294355708, 6.16604612815841615831329739324, 8.195560449330679294050428180333, 9.116054750146157160720349170313, 11.11645722146823459492028849731, 12.33634553537274589819945151389, 13.35461183517321495834938142685, 14.86487374155000717722823793686

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