Properties

Degree 2
Conductor $ 2^{2} \cdot 3 $
Sign $-0.354 - 0.934i$
Motivic weight 19
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (641. + 336. i)2-s + (−3.30e4 − 8.17e3i)3-s + (2.98e5 + 4.31e5i)4-s − 2.18e6i·5-s + (−1.84e7 − 1.63e7i)6-s − 4.59e7i·7-s + (4.61e7 + 3.76e8i)8-s + (1.02e9 + 5.41e8i)9-s + (7.34e8 − 1.40e9i)10-s + 1.63e8·11-s + (−6.34e9 − 1.67e10i)12-s + 2.15e9·13-s + (1.54e10 − 2.94e10i)14-s + (−1.78e10 + 7.22e10i)15-s + (−9.71e10 + 2.57e11i)16-s + 3.11e11i·17-s + ⋯
L(s)  = 1  + (0.885 + 0.464i)2-s + (−0.970 − 0.239i)3-s + (0.568 + 0.822i)4-s − 0.500i·5-s + (−0.748 − 0.663i)6-s − 0.430i·7-s + (0.121 + 0.992i)8-s + (0.885 + 0.465i)9-s + (0.232 − 0.442i)10-s + 0.0208·11-s + (−0.354 − 0.934i)12-s + 0.0563·13-s + (0.200 − 0.381i)14-s + (−0.119 + 0.485i)15-s + (−0.353 + 0.935i)16-s + 0.637i·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.354 - 0.934i)\, \overline{\Lambda}(20-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & (-0.354 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(12\)    =    \(2^{2} \cdot 3\)
\( \varepsilon \)  =  $-0.354 - 0.934i$
motivic weight  =  \(19\)
character  :  $\chi_{12} (11, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 12,\ (\ :19/2),\ -0.354 - 0.934i)$
$L(10)$  $\approx$  $1.895970983$
$L(\frac12)$  $\approx$  $1.895970983$
$L(\frac{21}{2})$   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 \( 1 + (-641. - 336. i)T \)
3 \( 1 + (3.30e4 + 8.17e3i)T \)
good5 \( 1 + 2.18e6iT - 1.90e13T^{2} \)
7 \( 1 + 4.59e7iT - 1.13e16T^{2} \)
11 \( 1 - 1.63e8T + 6.11e19T^{2} \)
13 \( 1 - 2.15e9T + 1.46e21T^{2} \)
17 \( 1 - 3.11e11iT - 2.39e23T^{2} \)
19 \( 1 - 2.29e12iT - 1.97e24T^{2} \)
23 \( 1 + 1.23e13T + 7.46e25T^{2} \)
29 \( 1 - 1.30e14iT - 6.10e27T^{2} \)
31 \( 1 - 8.17e13iT - 2.16e28T^{2} \)
37 \( 1 - 2.92e14T + 6.24e29T^{2} \)
41 \( 1 - 2.15e15iT - 4.39e30T^{2} \)
43 \( 1 + 6.38e14iT - 1.08e31T^{2} \)
47 \( 1 + 5.11e15T + 5.88e31T^{2} \)
53 \( 1 + 5.34e15iT - 5.77e32T^{2} \)
59 \( 1 - 5.99e16T + 4.42e33T^{2} \)
61 \( 1 - 8.58e16T + 8.34e33T^{2} \)
67 \( 1 + 3.62e17iT - 4.95e34T^{2} \)
71 \( 1 + 4.95e17T + 1.49e35T^{2} \)
73 \( 1 - 7.06e17T + 2.53e35T^{2} \)
79 \( 1 - 1.01e18iT - 1.13e36T^{2} \)
83 \( 1 - 4.60e17T + 2.90e36T^{2} \)
89 \( 1 + 5.30e18iT - 1.09e37T^{2} \)
97 \( 1 + 1.13e19T + 5.60e37T^{2} \)
show more
show less
\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−16.15702826643277563090306090229, −14.40498633563211070384157115866, −12.92625466335436548720870197208, −12.05102353943400951939623650672, −10.52555168474190026621714579581, −8.022449489984422470424396854352, −6.52447453974428406917937107078, −5.28767666093449495151091596103, −3.92129757214929540749694680787, −1.51325095012027008065195311829, 0.52318576347219450569583596991, 2.46672252285726233017493907828, 4.25436592158300375510711698169, 5.63201833081107026132587946228, 6.87287941281550445360855805127, 9.754504245157762219191535612781, 11.09488520998634104791923999347, 11.99731351318441068861968222037, 13.45872048068475367171253381708, 15.06893529887468118807847794028

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