Properties

Degree 2
Conductor $ 2^{2} \cdot 3 $
Sign $-0.949 + 0.313i$
Motivic weight 19
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−183. + 700. i)2-s + (2.29e4 − 2.51e4i)3-s + (−4.57e5 − 2.56e5i)4-s − 6.84e6i·5-s + (1.34e7 + 2.07e7i)6-s − 8.10e7i·7-s + (2.63e8 − 2.73e8i)8-s + (−1.05e8 − 1.15e9i)9-s + (4.79e9 + 1.25e9i)10-s − 1.02e10·11-s + (−1.69e10 + 5.61e9i)12-s − 9.13e9·13-s + (5.67e10 + 1.48e10i)14-s + (−1.72e11 − 1.57e11i)15-s + (1.43e11 + 2.34e11i)16-s + 6.29e11i·17-s + ⋯
L(s)  = 1  + (−0.253 + 0.967i)2-s + (0.674 − 0.738i)3-s + (−0.871 − 0.489i)4-s − 1.56i·5-s + (0.544 + 0.839i)6-s − 0.759i·7-s + (0.694 − 0.719i)8-s + (−0.0911 − 0.995i)9-s + (1.51 + 0.396i)10-s − 1.31·11-s + (−0.949 + 0.313i)12-s − 0.238·13-s + (0.734 + 0.192i)14-s + (−1.15 − 1.05i)15-s + (0.520 + 0.853i)16-s + 1.28i·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.949 + 0.313i)\, \overline{\Lambda}(20-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & (-0.949 + 0.313i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(12\)    =    \(2^{2} \cdot 3\)
\( \varepsilon \)  =  $-0.949 + 0.313i$
motivic weight  =  \(19\)
character  :  $\chi_{12} (11, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 12,\ (\ :19/2),\ -0.949 + 0.313i)$
$L(10)$  $\approx$  $0.8747865511$
$L(\frac12)$  $\approx$  $0.8747865511$
$L(\frac{21}{2})$   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 \( 1 + (183. - 700. i)T \)
3 \( 1 + (-2.29e4 + 2.51e4i)T \)
good5 \( 1 + 6.84e6iT - 1.90e13T^{2} \)
7 \( 1 + 8.10e7iT - 1.13e16T^{2} \)
11 \( 1 + 1.02e10T + 6.11e19T^{2} \)
13 \( 1 + 9.13e9T + 1.46e21T^{2} \)
17 \( 1 - 6.29e11iT - 2.39e23T^{2} \)
19 \( 1 - 2.01e12iT - 1.97e24T^{2} \)
23 \( 1 - 2.86e12T + 7.46e25T^{2} \)
29 \( 1 + 8.26e12iT - 6.10e27T^{2} \)
31 \( 1 + 1.75e14iT - 2.16e28T^{2} \)
37 \( 1 - 1.30e15T + 6.24e29T^{2} \)
41 \( 1 - 5.00e14iT - 4.39e30T^{2} \)
43 \( 1 + 2.29e15iT - 1.08e31T^{2} \)
47 \( 1 + 8.03e15T + 5.88e31T^{2} \)
53 \( 1 + 2.50e16iT - 5.77e32T^{2} \)
59 \( 1 + 6.40e16T + 4.42e33T^{2} \)
61 \( 1 - 4.18e16T + 8.34e33T^{2} \)
67 \( 1 - 8.57e16iT - 4.95e34T^{2} \)
71 \( 1 + 5.20e17T + 1.49e35T^{2} \)
73 \( 1 + 3.77e17T + 2.53e35T^{2} \)
79 \( 1 - 4.09e17iT - 1.13e36T^{2} \)
83 \( 1 + 1.71e18T + 2.90e36T^{2} \)
89 \( 1 + 6.97e17iT - 1.09e37T^{2} \)
97 \( 1 - 8.71e18T + 5.60e37T^{2} \)
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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

−14.87737956491278732546810718167, −13.35269340641354817661046602740, −12.74729180864096415600808510102, −9.898191365014031085243890060891, −8.392195860599216578426241986774, −7.71427223949663344686200697858, −5.79878501670451172115219316211, −4.18681495604874361039555596390, −1.47803592351176103812162292829, −0.28430807578795775144015459866, 2.67254409955910242170059679304, 2.84249413710653697592918922949, 4.92397305339622730745796954442, 7.55167620489062109197723543171, 9.186800147218112816042326742169, 10.40519172764212707761185103660, 11.34790509359225718388454453209, 13.36758959487537327523983794556, 14.61204294013077181133898426757, 15.79574394942778231198314060562

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