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Properties

Label 2-12-12.11-c17-0-22
Degree $2$
Conductor $12$
Sign $-0.987 + 0.160i$
Analytic cond. $21.9866$
Root an. cond. $4.68899$
Motivic weight $17$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−296. − 207. i)2-s + (2.15e3 − 1.11e4i)3-s + (4.51e4 + 1.23e5i)4-s + 1.59e6i·5-s + (−2.95e6 + 2.86e6i)6-s + 3.40e6i·7-s + (1.20e7 − 4.58e7i)8-s + (−1.19e8 − 4.81e7i)9-s + (3.30e8 − 4.73e8i)10-s + 4.20e8·11-s + (1.47e9 − 2.38e8i)12-s − 2.89e9·13-s + (7.05e8 − 1.01e9i)14-s + (1.77e10 + 3.44e9i)15-s + (−1.31e10 + 1.11e10i)16-s − 2.65e10i·17-s + ⋯
L(s)  = 1  + (−0.819 − 0.572i)2-s + (0.189 − 0.981i)3-s + (0.344 + 0.938i)4-s + 1.82i·5-s + (−0.717 + 0.696i)6-s + 0.223i·7-s + (0.254 − 0.966i)8-s + (−0.927 − 0.372i)9-s + (1.04 − 1.49i)10-s + 0.591·11-s + (0.987 − 0.160i)12-s − 0.985·13-s + (0.127 − 0.183i)14-s + (1.79 + 0.346i)15-s + (−0.762 + 0.646i)16-s − 0.924i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 + 0.160i)\, \overline{\Lambda}(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & (-0.987 + 0.160i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $-0.987 + 0.160i$
Analytic conductor: \(21.9866\)
Root analytic conductor: \(4.68899\)
Motivic weight: \(17\)
Rational: no
Arithmetic: yes
Character: $\chi_{12} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 12,\ (\ :17/2),\ -0.987 + 0.160i)\)

Particular Values

\(L(9)\) \(\approx\) \(0.0333144 - 0.413616i\)
\(L(\frac12)\) \(\approx\) \(0.0333144 - 0.413616i\)
\(L(\frac{19}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (296. + 207. i)T \)
3 \( 1 + (-2.15e3 + 1.11e4i)T \)
good5 \( 1 - 1.59e6iT - 7.62e11T^{2} \)
7 \( 1 - 3.40e6iT - 2.32e14T^{2} \)
11 \( 1 - 4.20e8T + 5.05e17T^{2} \)
13 \( 1 + 2.89e9T + 8.65e18T^{2} \)
17 \( 1 + 2.65e10iT - 8.27e20T^{2} \)
19 \( 1 + 1.07e11iT - 5.48e21T^{2} \)
23 \( 1 - 1.73e10T + 1.41e23T^{2} \)
29 \( 1 - 2.16e12iT - 7.25e24T^{2} \)
31 \( 1 + 6.70e12iT - 2.25e25T^{2} \)
37 \( 1 + 1.76e13T + 4.56e26T^{2} \)
41 \( 1 + 1.33e13iT - 2.61e27T^{2} \)
43 \( 1 + 1.66e13iT - 5.87e27T^{2} \)
47 \( 1 + 2.06e13T + 2.66e28T^{2} \)
53 \( 1 + 4.56e14iT - 2.05e29T^{2} \)
59 \( 1 + 2.10e15T + 1.27e30T^{2} \)
61 \( 1 + 6.78e14T + 2.24e30T^{2} \)
67 \( 1 + 4.32e15iT - 1.10e31T^{2} \)
71 \( 1 - 3.83e15T + 2.96e31T^{2} \)
73 \( 1 + 3.10e15T + 4.74e31T^{2} \)
79 \( 1 + 1.62e16iT - 1.81e32T^{2} \)
83 \( 1 + 1.61e16T + 4.21e32T^{2} \)
89 \( 1 - 4.61e16iT - 1.37e33T^{2} \)
97 \( 1 + 4.10e16T + 5.95e33T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−15.11321757902218946468395435064, −13.76245451147600694514704121662, −11.99199838568310029575722390100, −11.00071009325388116339730025161, −9.357521099725031009898303485421, −7.45571850647558796747669720082, −6.71392139343936917861590235012, −3.07970635396073497755913072135, −2.19574112715782467422484380200, −0.18413387784789912276949691429, 1.45952945676638169873047283521, 4.36958143728807594143724641050, 5.62804338658008885636266597188, 8.101698344343895956170972554476, 9.102180496889161048135320462593, 10.19176640177940484112122254780, 12.15371023474209404476341615492, 14.20552536016729853119299010590, 15.60170583550950271064575349401, 16.78104449979941915751852823991

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