Properties

Label 2-12-12.11-c17-0-11
Degree $2$
Conductor $12$
Sign $-0.498 - 0.866i$
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  + (−267. + 244. i)2-s + (−9.29e3 + 6.54e3i)3-s + (1.19e4 − 1.30e5i)4-s + 1.19e6i·5-s + (8.88e5 − 4.01e6i)6-s − 5.99e6i·7-s + (2.86e7 + 3.78e7i)8-s + (4.35e7 − 1.21e8i)9-s + (−2.92e8 − 3.20e8i)10-s + 1.37e9·11-s + (7.42e8 + 1.29e9i)12-s + 3.90e9·13-s + (1.46e9 + 1.60e9i)14-s + (−7.83e9 − 1.11e10i)15-s + (−1.68e10 − 3.12e9i)16-s + 1.32e10i·17-s + ⋯
L(s)  = 1  + (−0.738 + 0.674i)2-s + (−0.817 + 0.575i)3-s + (0.0912 − 0.995i)4-s + 1.37i·5-s + (0.216 − 0.976i)6-s − 0.392i·7-s + (0.603 + 0.797i)8-s + (0.337 − 0.941i)9-s + (−0.924 − 1.01i)10-s + 1.92·11-s + (0.498 + 0.866i)12-s + 1.32·13-s + (0.264 + 0.290i)14-s + (−0.789 − 1.12i)15-s + (−0.983 − 0.181i)16-s + 0.462i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.498 - 0.866i)\, \overline{\Lambda}(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & (-0.498 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $-0.498 - 0.866i$
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: \((2,\ 12,\ (\ :17/2),\ -0.498 - 0.866i)\)

Particular Values

\(L(9)\) \(\approx\) \(0.573375 + 0.991198i\)
\(L(\frac12)\) \(\approx\) \(0.573375 + 0.991198i\)
\(L(\frac{19}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (267. - 244. i)T \)
3 \( 1 + (9.29e3 - 6.54e3i)T \)
good5 \( 1 - 1.19e6iT - 7.62e11T^{2} \)
7 \( 1 + 5.99e6iT - 2.32e14T^{2} \)
11 \( 1 - 1.37e9T + 5.05e17T^{2} \)
13 \( 1 - 3.90e9T + 8.65e18T^{2} \)
17 \( 1 - 1.32e10iT - 8.27e20T^{2} \)
19 \( 1 + 4.75e10iT - 5.48e21T^{2} \)
23 \( 1 + 1.03e11T + 1.41e23T^{2} \)
29 \( 1 - 3.56e12iT - 7.25e24T^{2} \)
31 \( 1 + 1.33e12iT - 2.25e25T^{2} \)
37 \( 1 - 3.10e13T + 4.56e26T^{2} \)
41 \( 1 + 6.07e13iT - 2.61e27T^{2} \)
43 \( 1 - 2.79e12iT - 5.87e27T^{2} \)
47 \( 1 + 6.60e13T + 2.66e28T^{2} \)
53 \( 1 - 2.50e14iT - 2.05e29T^{2} \)
59 \( 1 + 6.76e14T + 1.27e30T^{2} \)
61 \( 1 - 5.37e14T + 2.24e30T^{2} \)
67 \( 1 - 7.52e14iT - 1.10e31T^{2} \)
71 \( 1 + 4.09e15T + 2.96e31T^{2} \)
73 \( 1 - 6.58e15T + 4.74e31T^{2} \)
79 \( 1 - 1.47e16iT - 1.81e32T^{2} \)
83 \( 1 - 1.45e16T + 4.21e32T^{2} \)
89 \( 1 + 2.31e16iT - 1.37e33T^{2} \)
97 \( 1 + 5.89e16T + 5.95e33T^{2} \)
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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

−16.52747338605348148892810999930, −15.18768241807214319658495214926, −14.18825051440068045106253776012, −11.37216302435705052104602731663, −10.58243316358065995993241868390, −9.136379744774334320350491879703, −6.89557683261601444920048757254, −6.13890529075449120876444446188, −3.88954583319585009488827512907, −1.10856842834752453367359872328, 0.789800651997020642275200391112, 1.55708281828329527738297471467, 4.21077529132102127102963471443, 6.24020130340441702964766036772, 8.231006318622727219052008758168, 9.422259802468884148996642379501, 11.44173501193375460219900715959, 12.19222051722365005232243164755, 13.40655509222865129997223132680, 16.25893967925062124106109227870

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