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Properties

Label 2-12-12.11-c17-0-6
Degree $2$
Conductor $12$
Sign $-0.422 + 0.906i$
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  + (207. + 296. i)2-s + (−1.13e4 − 959. i)3-s + (−4.51e4 + 1.23e5i)4-s + 8.04e5i·5-s + (−2.06e6 − 3.55e6i)6-s + 2.45e7i·7-s + (−4.58e7 + 1.21e7i)8-s + (1.27e8 + 2.17e7i)9-s + (−2.38e8 + 1.66e8i)10-s + 7.53e8·11-s + (6.28e8 − 1.35e9i)12-s − 5.41e9·13-s + (−7.27e9 + 5.08e9i)14-s + (7.71e8 − 9.11e9i)15-s + (−1.31e10 − 1.11e10i)16-s − 2.92e10i·17-s + ⋯
L(s)  = 1  + (0.572 + 0.819i)2-s + (−0.996 − 0.0844i)3-s + (−0.344 + 0.938i)4-s + 0.921i·5-s + (−0.501 − 0.865i)6-s + 1.60i·7-s + (−0.966 + 0.255i)8-s + (0.985 + 0.168i)9-s + (−0.755 + 0.527i)10-s + 1.05·11-s + (0.422 − 0.906i)12-s − 1.84·13-s + (−1.31 + 0.920i)14-s + (0.0777 − 0.917i)15-s + (−0.763 − 0.646i)16-s − 1.01i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.422 + 0.906i)\, \overline{\Lambda}(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & (-0.422 + 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $-0.422 + 0.906i$
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: \((2,\ 12,\ (\ :17/2),\ -0.422 + 0.906i)\)

Particular Values

\(L(9)\) \(\approx\) \(0.527529 - 0.827586i\)
\(L(\frac12)\) \(\approx\) \(0.527529 - 0.827586i\)
\(L(\frac{19}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-207. - 296. i)T \)
3 \( 1 + (1.13e4 + 959. i)T \)
good5 \( 1 - 8.04e5iT - 7.62e11T^{2} \)
7 \( 1 - 2.45e7iT - 2.32e14T^{2} \)
11 \( 1 - 7.53e8T + 5.05e17T^{2} \)
13 \( 1 + 5.41e9T + 8.65e18T^{2} \)
17 \( 1 + 2.92e10iT - 8.27e20T^{2} \)
19 \( 1 - 4.12e10iT - 5.48e21T^{2} \)
23 \( 1 - 2.06e11T + 1.41e23T^{2} \)
29 \( 1 + 2.39e12iT - 7.25e24T^{2} \)
31 \( 1 - 2.92e12iT - 2.25e25T^{2} \)
37 \( 1 - 8.68e12T + 4.56e26T^{2} \)
41 \( 1 + 1.59e13iT - 2.61e27T^{2} \)
43 \( 1 - 7.46e13iT - 5.87e27T^{2} \)
47 \( 1 - 5.30e12T + 2.66e28T^{2} \)
53 \( 1 - 5.59e14iT - 2.05e29T^{2} \)
59 \( 1 - 8.13e13T + 1.27e30T^{2} \)
61 \( 1 + 1.52e15T + 2.24e30T^{2} \)
67 \( 1 - 3.40e14iT - 1.10e31T^{2} \)
71 \( 1 - 2.49e15T + 2.96e31T^{2} \)
73 \( 1 + 3.35e15T + 4.74e31T^{2} \)
79 \( 1 + 7.33e15iT - 1.81e32T^{2} \)
83 \( 1 - 2.82e14T + 4.21e32T^{2} \)
89 \( 1 + 9.20e15iT - 1.37e33T^{2} \)
97 \( 1 + 3.56e16T + 5.95e33T^{2} \)
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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

−16.77275659167244879271895905695, −15.35363528961585653583061573420, −14.44998399261890688551130780541, −12.38310275094513107477522093769, −11.68141090409206227276550282064, −9.437066723357388528436293048667, −7.26606898978760551956567147647, −6.12414569203057785384080505992, −4.86138507405124523924642434585, −2.66567123760257525627251292323, 0.34575206284559571003513506267, 1.34825662965146040876309649970, 4.07071346185515740488744000875, 4.98124751164427449561748034128, 6.87563630062972834616622293926, 9.584631776185047947625113708062, 10.77141604264469804643704718118, 12.14054798601205951919163564997, 13.12351776289943046893950201363, 14.70488139693013622873841819388

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