Properties

Degree 2
Conductor $ 2^{2} \cdot 3 $
Sign $-0.985 - 0.170i$
Motivic weight 15
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (80.5 − 162. i)2-s + (2.76e3 − 2.58e3i)3-s + (−1.97e4 − 2.61e4i)4-s − 3.32e5i·5-s + (−1.96e5 − 6.57e5i)6-s + 2.13e6i·7-s + (−5.82e6 + 1.09e6i)8-s + (9.67e5 − 1.43e7i)9-s + (−5.39e7 − 2.68e7i)10-s + 7.24e7·11-s + (−1.22e8 − 2.11e7i)12-s + 5.97e7·13-s + (3.46e8 + 1.72e8i)14-s + (−8.60e8 − 9.20e8i)15-s + (−2.91e8 + 1.03e9i)16-s + 9.88e8i·17-s + ⋯
L(s)  = 1  + (0.445 − 0.895i)2-s + (0.730 − 0.682i)3-s + (−0.603 − 0.797i)4-s − 1.90i·5-s + (−0.286 − 0.958i)6-s + 0.979i·7-s + (−0.982 + 0.185i)8-s + (0.0674 − 0.997i)9-s + (−1.70 − 0.847i)10-s + 1.12·11-s + (−0.985 − 0.170i)12-s + 0.264·13-s + (0.877 + 0.436i)14-s + (−1.30 − 1.39i)15-s + (−0.271 + 0.962i)16-s + 0.584i·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 - 0.170i)\, \overline{\Lambda}(16-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & (-0.985 - 0.170i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(12\)    =    \(2^{2} \cdot 3\)
\( \varepsilon \)  =  $-0.985 - 0.170i$
motivic weight  =  \(15\)
character  :  $\chi_{12} (11, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 12,\ (\ :15/2),\ -0.985 - 0.170i)$
$L(8)$  $\approx$  $0.228375 + 2.66086i$
$L(\frac12)$  $\approx$  $0.228375 + 2.66086i$
$L(\frac{17}{2})$   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 \( 1 + (-80.5 + 162. i)T \)
3 \( 1 + (-2.76e3 + 2.58e3i)T \)
good5 \( 1 + 3.32e5iT - 3.05e10T^{2} \)
7 \( 1 - 2.13e6iT - 4.74e12T^{2} \)
11 \( 1 - 7.24e7T + 4.17e15T^{2} \)
13 \( 1 - 5.97e7T + 5.11e16T^{2} \)
17 \( 1 - 9.88e8iT - 2.86e18T^{2} \)
19 \( 1 + 6.22e7iT - 1.51e19T^{2} \)
23 \( 1 - 2.88e9T + 2.66e20T^{2} \)
29 \( 1 - 3.09e10iT - 8.62e21T^{2} \)
31 \( 1 + 4.97e10iT - 2.34e22T^{2} \)
37 \( 1 - 2.95e11T + 3.33e23T^{2} \)
41 \( 1 + 1.49e12iT - 1.55e24T^{2} \)
43 \( 1 + 2.69e12iT - 3.17e24T^{2} \)
47 \( 1 - 4.45e12T + 1.20e25T^{2} \)
53 \( 1 - 3.63e12iT - 7.31e25T^{2} \)
59 \( 1 + 2.17e13T + 3.65e26T^{2} \)
61 \( 1 + 3.26e12T + 6.02e26T^{2} \)
67 \( 1 - 7.11e13iT - 2.46e27T^{2} \)
71 \( 1 - 4.18e13T + 5.87e27T^{2} \)
73 \( 1 - 1.11e14T + 8.90e27T^{2} \)
79 \( 1 + 1.98e14iT - 2.91e28T^{2} \)
83 \( 1 + 1.29e14T + 6.11e28T^{2} \)
89 \( 1 - 1.60e14iT - 1.74e29T^{2} \)
97 \( 1 + 1.03e15T + 6.33e29T^{2} \)
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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

−15.33585855234767112396250144007, −13.78866050627059083089253622980, −12.58615659491192388904057473675, −11.99698709105116712095391004676, −9.223192979358329566485589749080, −8.651928171554073564135601514770, −5.73462320251582310792721489580, −4.00865695449447753694982244641, −1.94803837266857507645134763871, −0.870567588653024221607501967607, 3.03556287492457910144864646842, 4.10526212406021671012584967932, 6.53607214666335500257099234593, 7.65899915038901792152188920961, 9.644393327690475565604570854649, 11.13410709047506624116070102302, 13.77749965715620016558103969482, 14.38069369699050546359652972463, 15.36948483897217420010560431586, 16.83307996303808121763309719337

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