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Properties

Label 2-12-12.11-c17-0-19
Degree $2$
Conductor $12$
Sign $0.950 + 0.309i$
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  + (−232. + 277. i)2-s + (5.37e3 − 1.00e4i)3-s + (−2.31e4 − 1.29e5i)4-s − 2.93e5i·5-s + (1.53e6 + 3.81e6i)6-s + 2.56e7i·7-s + (4.12e7 + 2.35e7i)8-s + (−7.13e7 − 1.07e8i)9-s + (8.15e7 + 6.82e7i)10-s + 2.08e8·11-s + (−1.41e9 − 4.61e8i)12-s + 1.77e9·13-s + (−7.11e9 − 5.94e9i)14-s + (−2.94e9 − 1.57e9i)15-s + (−1.61e10 + 5.98e9i)16-s − 1.71e10i·17-s + ⋯
L(s)  = 1  + (−0.641 + 0.767i)2-s + (0.472 − 0.881i)3-s + (−0.176 − 0.984i)4-s − 0.336i·5-s + (0.372 + 0.928i)6-s + 1.67i·7-s + (0.868 + 0.495i)8-s + (−0.552 − 0.833i)9-s + (0.258 + 0.215i)10-s + 0.293·11-s + (−0.950 − 0.309i)12-s + 0.604·13-s + (−1.28 − 1.07i)14-s + (−0.296 − 0.159i)15-s + (−0.937 + 0.348i)16-s − 0.595i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.950 + 0.309i)\, \overline{\Lambda}(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & (0.950 + 0.309i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $0.950 + 0.309i$
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: \((2,\ 12,\ (\ :17/2),\ 0.950 + 0.309i)\)

Particular Values

\(L(9)\) \(\approx\) \(1.58112 - 0.250880i\)
\(L(\frac12)\) \(\approx\) \(1.58112 - 0.250880i\)
\(L(\frac{19}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (232. - 277. i)T \)
3 \( 1 + (-5.37e3 + 1.00e4i)T \)
good5 \( 1 + 2.93e5iT - 7.62e11T^{2} \)
7 \( 1 - 2.56e7iT - 2.32e14T^{2} \)
11 \( 1 - 2.08e8T + 5.05e17T^{2} \)
13 \( 1 - 1.77e9T + 8.65e18T^{2} \)
17 \( 1 + 1.71e10iT - 8.27e20T^{2} \)
19 \( 1 + 4.51e10iT - 5.48e21T^{2} \)
23 \( 1 - 5.42e11T + 1.41e23T^{2} \)
29 \( 1 + 4.58e12iT - 7.25e24T^{2} \)
31 \( 1 - 3.35e12iT - 2.25e25T^{2} \)
37 \( 1 - 2.35e12T + 4.56e26T^{2} \)
41 \( 1 - 5.23e12iT - 2.61e27T^{2} \)
43 \( 1 + 6.12e13iT - 5.87e27T^{2} \)
47 \( 1 - 2.77e14T + 2.66e28T^{2} \)
53 \( 1 + 8.52e14iT - 2.05e29T^{2} \)
59 \( 1 - 1.48e15T + 1.27e30T^{2} \)
61 \( 1 - 1.60e15T + 2.24e30T^{2} \)
67 \( 1 - 1.40e14iT - 1.10e31T^{2} \)
71 \( 1 + 3.73e15T + 2.96e31T^{2} \)
73 \( 1 + 8.49e15T + 4.74e31T^{2} \)
79 \( 1 - 1.88e16iT - 1.81e32T^{2} \)
83 \( 1 + 1.14e15T + 4.21e32T^{2} \)
89 \( 1 - 1.71e16iT - 1.37e33T^{2} \)
97 \( 1 - 1.18e17T + 5.95e33T^{2} \)
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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

−15.74911823083304097824652966363, −14.71346542560728970187772686943, −13.18650450128913139307429487474, −11.63648201810156242069261444901, −9.170262882858430911503915151909, −8.497673283994888291923592142266, −6.80072219005223429419349826991, −5.44123948414926890378408861517, −2.41442243404509319100983816895, −0.834381233760649221426312993082, 1.13598445732756819879500109797, 3.21608567290690266188315182712, 4.25558081429059081759718338475, 7.31277103704791913405518957751, 8.853617322429079089778963257149, 10.36193638778580169469186054053, 10.96884342525308881622010455438, 13.18396438261377512694208012247, 14.46805148335914404366125730033, 16.37879312854063645219884435912

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