Properties

Degree 2
Conductor $ 2^{2} \cdot 3 $
Sign $0.871 - 0.489i$
Motivic weight 15
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (171. − 57.9i)2-s + (3.75e3 + 526. i)3-s + (2.60e4 − 1.98e4i)4-s + 1.72e5i·5-s + (6.73e5 − 1.26e5i)6-s + 3.89e6i·7-s + (3.31e6 − 4.91e6i)8-s + (1.37e7 + 3.95e6i)9-s + (9.99e6 + 2.95e7i)10-s − 4.37e7·11-s + (1.08e8 − 6.07e7i)12-s − 4.08e7·13-s + (2.25e8 + 6.67e8i)14-s + (−9.08e7 + 6.47e8i)15-s + (2.84e8 − 1.03e9i)16-s − 1.04e9i·17-s + ⋯
L(s)  = 1  + (0.947 − 0.319i)2-s + (0.990 + 0.139i)3-s + (0.795 − 0.606i)4-s + 0.987i·5-s + (0.982 − 0.185i)6-s + 1.78i·7-s + (0.559 − 0.828i)8-s + (0.961 + 0.275i)9-s + (0.315 + 0.935i)10-s − 0.677·11-s + (0.871 − 0.489i)12-s − 0.180·13-s + (0.571 + 1.69i)14-s + (−0.137 + 0.978i)15-s + (0.264 − 0.964i)16-s − 0.620i·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.871 - 0.489i)\, \overline{\Lambda}(16-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & (0.871 - 0.489i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(12\)    =    \(2^{2} \cdot 3\)
\( \varepsilon \)  =  $0.871 - 0.489i$
motivic weight  =  \(15\)
character  :  $\chi_{12} (11, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 12,\ (\ :15/2),\ 0.871 - 0.489i)$
$L(8)$  $\approx$  $4.42592 + 1.15818i$
$L(\frac12)$  $\approx$  $4.42592 + 1.15818i$
$L(\frac{17}{2})$   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 \( 1 + (-171. + 57.9i)T \)
3 \( 1 + (-3.75e3 - 526. i)T \)
good5 \( 1 - 1.72e5iT - 3.05e10T^{2} \)
7 \( 1 - 3.89e6iT - 4.74e12T^{2} \)
11 \( 1 + 4.37e7T + 4.17e15T^{2} \)
13 \( 1 + 4.08e7T + 5.11e16T^{2} \)
17 \( 1 + 1.04e9iT - 2.86e18T^{2} \)
19 \( 1 + 4.29e9iT - 1.51e19T^{2} \)
23 \( 1 - 2.27e10T + 2.66e20T^{2} \)
29 \( 1 - 1.81e10iT - 8.62e21T^{2} \)
31 \( 1 + 1.17e11iT - 2.34e22T^{2} \)
37 \( 1 + 1.02e12T + 3.33e23T^{2} \)
41 \( 1 + 1.37e12iT - 1.55e24T^{2} \)
43 \( 1 + 7.55e11iT - 3.17e24T^{2} \)
47 \( 1 - 2.91e12T + 1.20e25T^{2} \)
53 \( 1 - 1.31e13iT - 7.31e25T^{2} \)
59 \( 1 - 1.15e12T + 3.65e26T^{2} \)
61 \( 1 + 1.08e13T + 6.02e26T^{2} \)
67 \( 1 + 5.22e13iT - 2.46e27T^{2} \)
71 \( 1 + 2.62e12T + 5.87e27T^{2} \)
73 \( 1 - 3.29e13T + 8.90e27T^{2} \)
79 \( 1 - 1.56e14iT - 2.91e28T^{2} \)
83 \( 1 - 3.43e13T + 6.11e28T^{2} \)
89 \( 1 - 5.29e14iT - 1.74e29T^{2} \)
97 \( 1 + 9.09e14T + 6.33e29T^{2} \)
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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

−15.50234813478577072655679134065, −15.16517812359692820657534610041, −13.81275211877095290068787661265, −12.41296941708726962174505295518, −10.83321133559465343760981265717, −9.131970683846488298740932966277, −7.05369911125934950973018175724, −5.18416270792978445845947722782, −3.00054526668422061880362228624, −2.35744556643769502617638654431, 1.36116287774433329404231250940, 3.48620075536618856989850102078, 4.73605402776812944238677876056, 7.10369616395803000667904464233, 8.277634380543275838070171329972, 10.42656618016363344687935569829, 12.68161196589399475925435486248, 13.45837424303037382276196199031, 14.58795037707215403069578133112, 16.13446994837463886153896159291

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