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Properties

Label 2-12-12.11-c17-0-29
Degree $2$
Conductor $12$
Sign $-0.856 + 0.516i$
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 + (1.67e6 − 3.75e6i)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 + (1.27e9 − 7.69e8i)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 + (0.406 − 0.913i)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 + (0.856 − 0.516i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.856 + 0.516i)\, \overline{\Lambda}(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & (-0.856 + 0.516i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $-0.856 + 0.516i$
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: \((2,\ 12,\ (\ :17/2),\ -0.856 + 0.516i)\)

Particular Values

\(L(9)\) \(\approx\) \(0.343858 - 1.23600i\)
\(L(\frac12)\) \(\approx\) \(0.343858 - 1.23600i\)
\(L(\frac{19}{2})\) 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

−15.37054045128686711348319852891, −13.48450119125523038321293696932, −12.76491935916603808887740424716, −11.80577840714447268444438251146, −8.764867515454008618324361186504, −7.60577990146298938565863068957, −5.77071408700583698644332275028, −4.67446269484603111455138468600, −2.19630978604893032886703579074, −0.31987647015602893890237182334, 2.58384573885149713481437933694, 3.61785247558346451494903914035, 5.36191184465368044394840975500, 6.97102958895525891398855008316, 10.01135969862495533592637452130, 10.63481932022393356645304031868, 11.91029530785967688582859597754, 14.13073147212715582741542021828, 14.75502070642852405630414936232, 16.00144104822863799922501619707

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