Properties

Degree 2
Conductor $ 2^{2} \cdot 3 $
Sign $0.749 - 0.662i$
Motivic weight 19
Primitive yes
Self-dual no
Analytic rank 0

Origins

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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 + (−2.39e7 + 5.88e6i)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 + (1.33e10 − 1.18e10i)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 + (−0.971 + 0.238i)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 + (0.749 − 0.662i)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

\[\begin{aligned} \Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 - 0.662i)\, \overline{\Lambda}(20-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & (0.749 - 0.662i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(12\)    =    \(2^{2} \cdot 3\)
\( \varepsilon \)  =  $0.749 - 0.662i$
motivic weight  =  \(19\)
character  :  $\chi_{12} (11, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 12,\ (\ :19/2),\ 0.749 - 0.662i)$
$L(10)$  $\approx$  $1.952226786$
$L(\frac12)$  $\approx$  $1.952226786$
$L(\frac{21}{2})$   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 \( 1 + 1.13e19T + 5.60e37T^{2} \)
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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

−15.60804129167451829762307145940, −14.68081643430382863938570095773, −13.00675902520757832499883377487, −10.89334830676763009387292078312, −9.282811335698245820417999179452, −8.548878774994215032722333915747, −7.01250667252821552871591470359, −4.97363756992413515762836186796, −2.69700148635361074947018226612, −1.11864681373031025057874484042, 0.962540227033930053264245070120, 2.45896183478198786224708014895, 3.71798234469545438789160639588, 6.88624661573290436712215423002, 8.018131335234778668028657895872, 9.433467969588990413750317308311, 10.67663664253811310342844029125, 12.38092793805582034905454591925, 13.86157529098694257270081229111, 15.33448123967104000659920971456

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