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Properties

Label 2-12-12.11-c15-0-10
Degree $2$
Conductor $12$
Sign $-0.259 - 0.965i$
Analytic cond. $17.1232$
Root an. cond. $4.13802$
Motivic weight $15$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−83.3 + 160. i)2-s + (3.55e3 + 1.30e3i)3-s + (−1.88e4 − 2.67e4i)4-s − 9.33e4i·5-s + (−5.06e5 + 4.62e5i)6-s + 1.14e6i·7-s + (5.87e6 − 8.04e5i)8-s + (1.09e7 + 9.29e6i)9-s + (1.50e7 + 7.77e6i)10-s − 1.80e6·11-s + (−3.21e7 − 1.19e8i)12-s + 3.17e8·13-s + (−1.84e8 − 9.54e7i)14-s + (1.21e8 − 3.31e8i)15-s + (−3.60e8 + 1.01e9i)16-s + 2.67e9i·17-s + ⋯
L(s)  = 1  + (−0.460 + 0.887i)2-s + (0.938 + 0.344i)3-s + (−0.576 − 0.817i)4-s − 0.534i·5-s + (−0.738 + 0.674i)6-s + 0.525i·7-s + (0.990 − 0.135i)8-s + (0.761 + 0.647i)9-s + (0.474 + 0.245i)10-s − 0.0279·11-s + (−0.259 − 0.965i)12-s + 1.40·13-s + (−0.466 − 0.241i)14-s + (0.184 − 0.501i)15-s + (−0.335 + 0.942i)16-s + 1.58i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.259 - 0.965i)\, \overline{\Lambda}(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & (-0.259 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $-0.259 - 0.965i$
Analytic conductor: \(17.1232\)
Root analytic conductor: \(4.13802\)
Motivic weight: \(15\)
Rational: no
Arithmetic: yes
Character: $\chi_{12} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 12,\ (\ :15/2),\ -0.259 - 0.965i)\)

Particular Values

\(L(8)\) \(\approx\) \(1.19239 + 1.55440i\)
\(L(\frac12)\) \(\approx\) \(1.19239 + 1.55440i\)
\(L(\frac{17}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (83.3 - 160. i)T \)
3 \( 1 + (-3.55e3 - 1.30e3i)T \)
good5 \( 1 + 9.33e4iT - 3.05e10T^{2} \)
7 \( 1 - 1.14e6iT - 4.74e12T^{2} \)
11 \( 1 + 1.80e6T + 4.17e15T^{2} \)
13 \( 1 - 3.17e8T + 5.11e16T^{2} \)
17 \( 1 - 2.67e9iT - 2.86e18T^{2} \)
19 \( 1 - 2.07e9iT - 1.51e19T^{2} \)
23 \( 1 + 2.43e10T + 2.66e20T^{2} \)
29 \( 1 + 4.77e9iT - 8.62e21T^{2} \)
31 \( 1 - 5.30e10iT - 2.34e22T^{2} \)
37 \( 1 - 4.06e11T + 3.33e23T^{2} \)
41 \( 1 + 1.36e12iT - 1.55e24T^{2} \)
43 \( 1 - 2.85e12iT - 3.17e24T^{2} \)
47 \( 1 - 7.27e11T + 1.20e25T^{2} \)
53 \( 1 - 7.05e12iT - 7.31e25T^{2} \)
59 \( 1 - 1.63e13T + 3.65e26T^{2} \)
61 \( 1 - 3.06e13T + 6.02e26T^{2} \)
67 \( 1 + 7.17e13iT - 2.46e27T^{2} \)
71 \( 1 + 4.82e13T + 5.87e27T^{2} \)
73 \( 1 + 9.56e13T + 8.90e27T^{2} \)
79 \( 1 + 3.27e14iT - 2.91e28T^{2} \)
83 \( 1 + 2.88e14T + 6.11e28T^{2} \)
89 \( 1 - 2.05e14iT - 1.74e29T^{2} \)
97 \( 1 + 5.41e14T + 6.33e29T^{2} \)
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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

−16.46799575764775024961337746027, −15.53487825056100928508153029160, −14.33463351731313654844104222185, −12.98675129788059349243203104789, −10.39393715228912876850286508039, −8.882169377924326417419410019879, −8.098726894377362884907577886635, −5.96362261840952752699082020687, −4.07386171954268901109676520929, −1.54290340883853369675145667251, 0.890608848914073938432205962272, 2.56312903475928462556854362709, 3.89750483836825611722319341047, 7.14757249812453561750161782706, 8.576066620427027227550780019621, 9.995197825059250275526389851025, 11.48599092157275339023325854274, 13.23288018008236079693944477251, 14.11436119694957867395744751357, 16.05768476529615689075724789321

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