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Properties

Label 2-12-12.11-c17-0-8
Degree $2$
Conductor $12$
Sign $-0.871 - 0.490i$
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  + (−361. + 22.3i)2-s + (9.13e3 + 6.75e3i)3-s + (1.30e5 − 1.61e4i)4-s + 8.41e5i·5-s + (−3.45e6 − 2.23e6i)6-s + 1.94e7i·7-s + (−4.66e7 + 8.74e6i)8-s + (3.79e7 + 1.23e8i)9-s + (−1.88e7 − 3.04e8i)10-s + 9.11e7·11-s + (1.29e9 + 7.30e8i)12-s + 3.49e9·13-s + (−4.34e8 − 7.02e9i)14-s + (−5.68e9 + 7.69e9i)15-s + (1.66e10 − 4.20e9i)16-s − 3.88e10i·17-s + ⋯
L(s)  = 1  + (−0.998 + 0.0617i)2-s + (0.804 + 0.594i)3-s + (0.992 − 0.123i)4-s + 0.963i·5-s + (−0.839 − 0.543i)6-s + 1.27i·7-s + (−0.982 + 0.184i)8-s + (0.293 + 0.955i)9-s + (−0.0595 − 0.961i)10-s + 0.128·11-s + (0.871 + 0.490i)12-s + 1.18·13-s + (−0.0786 − 1.27i)14-s + (−0.572 + 0.775i)15-s + (0.969 − 0.244i)16-s − 1.35i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.871 - 0.490i)\, \overline{\Lambda}(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & (-0.871 - 0.490i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $-0.871 - 0.490i$
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: \((2,\ 12,\ (\ :17/2),\ -0.871 - 0.490i)\)

Particular Values

\(L(9)\) \(\approx\) \(0.374442 + 1.42842i\)
\(L(\frac12)\) \(\approx\) \(0.374442 + 1.42842i\)
\(L(\frac{19}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (361. - 22.3i)T \)
3 \( 1 + (-9.13e3 - 6.75e3i)T \)
good5 \( 1 - 8.41e5iT - 7.62e11T^{2} \)
7 \( 1 - 1.94e7iT - 2.32e14T^{2} \)
11 \( 1 - 9.11e7T + 5.05e17T^{2} \)
13 \( 1 - 3.49e9T + 8.65e18T^{2} \)
17 \( 1 + 3.88e10iT - 8.27e20T^{2} \)
19 \( 1 - 1.11e11iT - 5.48e21T^{2} \)
23 \( 1 + 5.19e11T + 1.41e23T^{2} \)
29 \( 1 + 1.20e12iT - 7.25e24T^{2} \)
31 \( 1 + 2.70e12iT - 2.25e25T^{2} \)
37 \( 1 + 1.71e13T + 4.56e26T^{2} \)
41 \( 1 - 5.45e13iT - 2.61e27T^{2} \)
43 \( 1 + 3.09e13iT - 5.87e27T^{2} \)
47 \( 1 - 2.01e13T + 2.66e28T^{2} \)
53 \( 1 - 2.15e14iT - 2.05e29T^{2} \)
59 \( 1 + 5.19e14T + 1.27e30T^{2} \)
61 \( 1 + 2.32e15T + 2.24e30T^{2} \)
67 \( 1 - 3.25e15iT - 1.10e31T^{2} \)
71 \( 1 - 2.41e15T + 2.96e31T^{2} \)
73 \( 1 - 9.22e15T + 4.74e31T^{2} \)
79 \( 1 + 2.37e16iT - 1.81e32T^{2} \)
83 \( 1 - 2.02e16T + 4.21e32T^{2} \)
89 \( 1 + 6.58e16iT - 1.37e33T^{2} \)
97 \( 1 - 1.45e17T + 5.95e33T^{2} \)
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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

−16.22356628899667884160997517471, −15.35538602355928322585705823447, −14.19202418899883859373858715352, −11.75428365534622721540073656630, −10.34098239750832782856119608615, −9.115965487060885897782111689353, −7.88839347459651239152611646107, −6.03826848466008277358726344168, −3.24558417303622383088869216692, −2.03426171584698654463039814343, 0.65683118210792388486019369414, 1.62660859459026459922686430159, 3.73204222844077156611098043516, 6.62272018701476532560363316244, 8.056911974306594528012566467985, 9.031359649188341299099168029997, 10.68004195312417499268906034978, 12.52774625018101242706580805272, 13.77360998829724689723880266809, 15.62341684129601312754333001733

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