Properties

Label 2-6-3.2-c14-0-0
Degree $2$
Conductor $6$
Sign $-0.984 - 0.176i$
Analytic cond. $7.45973$
Root an. cond. $2.73125$
Motivic weight $14$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 90.5i·2-s + (385. − 2.15e3i)3-s − 8.19e3·4-s + 1.05e5i·5-s + (1.94e5 + 3.49e4i)6-s − 1.21e6·7-s − 7.41e5i·8-s + (−4.48e6 − 1.66e6i)9-s − 9.52e6·10-s + 1.71e7i·11-s + (−3.16e6 + 1.76e7i)12-s − 6.76e7·13-s − 1.10e8i·14-s + (2.26e8 + 4.06e7i)15-s + 6.71e7·16-s + 1.67e8i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.176 − 0.984i)3-s − 0.500·4-s + 1.34i·5-s + (0.696 + 0.124i)6-s − 1.47·7-s − 0.353i·8-s + (−0.937 − 0.347i)9-s − 0.952·10-s + 0.878i·11-s + (−0.0882 + 0.492i)12-s − 1.07·13-s − 1.04i·14-s + (1.32 + 0.237i)15-s + 0.250·16-s + 0.408i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 - 0.176i)\, \overline{\Lambda}(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & (-0.984 - 0.176i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6\)    =    \(2 \cdot 3\)
Sign: $-0.984 - 0.176i$
Analytic conductor: \(7.45973\)
Root analytic conductor: \(2.73125\)
Motivic weight: \(14\)
Rational: no
Arithmetic: yes
Character: $\chi_{6} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 6,\ (\ :7),\ -0.984 - 0.176i)\)

Particular Values

\(L(\frac{15}{2})\) \(\approx\) \(0.0485258 + 0.545860i\)
\(L(\frac12)\) \(\approx\) \(0.0485258 + 0.545860i\)
\(L(8)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 90.5iT \)
3 \( 1 + (-385. + 2.15e3i)T \)
good5 \( 1 - 1.05e5iT - 6.10e9T^{2} \)
7 \( 1 + 1.21e6T + 6.78e11T^{2} \)
11 \( 1 - 1.71e7iT - 3.79e14T^{2} \)
13 \( 1 + 6.76e7T + 3.93e15T^{2} \)
17 \( 1 - 1.67e8iT - 1.68e17T^{2} \)
19 \( 1 - 8.08e8T + 7.99e17T^{2} \)
23 \( 1 + 1.02e9iT - 1.15e19T^{2} \)
29 \( 1 + 1.38e10iT - 2.97e20T^{2} \)
31 \( 1 - 3.15e10T + 7.56e20T^{2} \)
37 \( 1 + 1.41e11T + 9.01e21T^{2} \)
41 \( 1 - 7.34e10iT - 3.79e22T^{2} \)
43 \( 1 + 1.15e11T + 7.38e22T^{2} \)
47 \( 1 - 6.63e11iT - 2.56e23T^{2} \)
53 \( 1 - 1.04e12iT - 1.37e24T^{2} \)
59 \( 1 + 2.50e11iT - 6.19e24T^{2} \)
61 \( 1 + 5.74e12T + 9.87e24T^{2} \)
67 \( 1 - 4.64e12T + 3.67e25T^{2} \)
71 \( 1 - 4.07e12iT - 8.27e25T^{2} \)
73 \( 1 - 4.29e12T + 1.22e26T^{2} \)
79 \( 1 + 1.78e13T + 3.68e26T^{2} \)
83 \( 1 - 5.35e13iT - 7.36e26T^{2} \)
89 \( 1 + 6.15e13iT - 1.95e27T^{2} \)
97 \( 1 - 1.58e12T + 6.52e27T^{2} \)
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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

−19.61545683337725517870678525818, −18.66535615593640327990938674099, −17.32395764857866381848401065896, −15.31396896184992671520771819989, −13.96589727846702875725784211184, −12.40353397594361885499732150717, −9.840631736238301368585436075544, −7.35713014484031204617586525010, −6.40564101039346302257844246107, −2.87767834652859316154557188946, 0.27283648421792362892319342421, 3.26918754319257457606715968709, 5.14067973023852954548439336995, 8.898499919286703670710916589200, 9.957376376338265255475525830888, 12.10351124382580211852189076219, 13.62456085068350837984457795917, 15.92405764183567679224416425982, 16.88222922564243206264219469883, 19.39845821353700139997847020356

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