Properties

Degree 2
Conductor $ 2^{2} \cdot 3 $
Sign $-0.983 - 0.182i$
Motivic weight 15
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (127. − 128. i)2-s + (692. − 3.72e3i)3-s + (−7.18 − 3.27e4i)4-s + 2.30e5i·5-s + (−3.88e5 − 5.65e5i)6-s − 3.60e6i·7-s + (−4.19e6 − 4.19e6i)8-s + (−1.33e7 − 5.15e6i)9-s + (2.95e7 + 2.95e7i)10-s − 5.00e7·11-s + (−1.22e8 − 2.26e7i)12-s + 3.24e7·13-s + (−4.61e8 − 4.61e8i)14-s + (8.58e8 + 1.59e8i)15-s + (−1.07e9 + 4.70e5i)16-s + 2.12e9i·17-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)2-s + (0.182 − 0.983i)3-s + (−0.000219 − 0.999i)4-s + 1.31i·5-s + (−0.566 − 0.824i)6-s − 1.65i·7-s + (−0.707 − 0.706i)8-s + (−0.933 − 0.359i)9-s + (0.933 + 0.933i)10-s − 0.773·11-s + (−0.983 − 0.182i)12-s + 0.143·13-s + (−1.17 − 1.17i)14-s + (1.29 + 0.241i)15-s + (−0.999 + 0.000438i)16-s + 1.25i·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 - 0.182i)\, \overline{\Lambda}(16-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & (-0.983 - 0.182i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(12\)    =    \(2^{2} \cdot 3\)
\( \varepsilon \)  =  $-0.983 - 0.182i$
motivic weight  =  \(15\)
character  :  $\chi_{12} (11, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 12,\ (\ :15/2),\ -0.983 - 0.182i)$
$L(8)$  $\approx$  $0.182073 + 1.97831i$
$L(\frac12)$  $\approx$  $0.182073 + 1.97831i$
$L(\frac{17}{2})$   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 \( 1 + (-127. + 128. i)T \)
3 \( 1 + (-692. + 3.72e3i)T \)
good5 \( 1 - 2.30e5iT - 3.05e10T^{2} \)
7 \( 1 + 3.60e6iT - 4.74e12T^{2} \)
11 \( 1 + 5.00e7T + 4.17e15T^{2} \)
13 \( 1 - 3.24e7T + 5.11e16T^{2} \)
17 \( 1 - 2.12e9iT - 2.86e18T^{2} \)
19 \( 1 + 2.21e9iT - 1.51e19T^{2} \)
23 \( 1 - 1.70e9T + 2.66e20T^{2} \)
29 \( 1 + 1.34e11iT - 8.62e21T^{2} \)
31 \( 1 + 1.31e11iT - 2.34e22T^{2} \)
37 \( 1 - 6.49e11T + 3.33e23T^{2} \)
41 \( 1 + 2.08e12iT - 1.55e24T^{2} \)
43 \( 1 + 9.33e9iT - 3.17e24T^{2} \)
47 \( 1 - 1.60e12T + 1.20e25T^{2} \)
53 \( 1 + 1.07e12iT - 7.31e25T^{2} \)
59 \( 1 - 1.36e13T + 3.65e26T^{2} \)
61 \( 1 - 2.10e13T + 6.02e26T^{2} \)
67 \( 1 + 6.54e13iT - 2.46e27T^{2} \)
71 \( 1 + 1.13e14T + 5.87e27T^{2} \)
73 \( 1 - 7.40e13T + 8.90e27T^{2} \)
79 \( 1 - 2.01e13iT - 2.91e28T^{2} \)
83 \( 1 - 1.31e14T + 6.11e28T^{2} \)
89 \( 1 + 4.75e13iT - 1.74e29T^{2} \)
97 \( 1 - 9.60e14T + 6.33e29T^{2} \)
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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

−15.02396178654267803836087408297, −13.88695693571011856125844180584, −13.08289370173861801116694343175, −11.22436363078956238842204777296, −10.32062014350884102559288691227, −7.50234304137262013524192446502, −6.28137637656671312844727775829, −3.76904088109053216377563424958, −2.32762023697316442688029873870, −0.56930173951345074840501258625, 2.82577364516946022539480798905, 4.82999903010261534705105319530, 5.55169490343049100272455118540, 8.332778316139712011564613525511, 9.212827881182427790886779578618, 11.76979820950113400932663720126, 12.95115743173714905592055080676, 14.64600820581289428683831441175, 15.89839685895723479750667016109, 16.36924007897877443408150400262

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