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Properties

Label 2-6-3.2-c22-0-4
Degree $2$
Conductor $6$
Sign $0.512 - 0.858i$
Analytic cond. $18.4024$
Root an. cond. $4.28980$
Motivic weight $22$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.44e3i·2-s + (1.52e5 + 9.07e4i)3-s − 2.09e6·4-s − 3.00e7i·5-s + (−1.31e8 + 2.20e8i)6-s + 3.33e9·7-s − 3.03e9i·8-s + (1.49e10 + 2.76e10i)9-s + 4.34e10·10-s − 3.94e11i·11-s + (−3.19e11 − 1.90e11i)12-s + 1.51e12·13-s + 4.82e12i·14-s + (2.72e12 − 4.56e12i)15-s + 4.39e12·16-s + 3.69e13i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.858 + 0.512i)3-s − 0.500·4-s − 0.614i·5-s + (−0.362 + 0.607i)6-s + 1.68·7-s − 0.353i·8-s + (0.475 + 0.879i)9-s + 0.434·10-s − 1.38i·11-s + (−0.429 − 0.256i)12-s + 0.847·13-s + 1.19i·14-s + (0.314 − 0.527i)15-s + 0.250·16-s + 1.07i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.512 - 0.858i)\, \overline{\Lambda}(23-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s+11) \, L(s)\cr =\mathstrut & (0.512 - 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6\)    =    \(2 \cdot 3\)
Sign: $0.512 - 0.858i$
Analytic conductor: \(18.4024\)
Root analytic conductor: \(4.28980\)
Motivic weight: \(22\)
Rational: no
Arithmetic: yes
Character: $\chi_{6} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 6,\ (\ :11),\ 0.512 - 0.858i)\)

Particular Values

\(L(\frac{23}{2})\) \(\approx\) \(2.55426 + 1.45079i\)
\(L(\frac12)\) \(\approx\) \(2.55426 + 1.45079i\)
\(L(12)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.44e3iT \)
3 \( 1 + (-1.52e5 - 9.07e4i)T \)
good5 \( 1 + 3.00e7iT - 2.38e15T^{2} \)
7 \( 1 - 3.33e9T + 3.90e18T^{2} \)
11 \( 1 + 3.94e11iT - 8.14e22T^{2} \)
13 \( 1 - 1.51e12T + 3.21e24T^{2} \)
17 \( 1 - 3.69e13iT - 1.17e27T^{2} \)
19 \( 1 + 3.98e13T + 1.35e28T^{2} \)
23 \( 1 - 1.32e15iT - 9.07e29T^{2} \)
29 \( 1 + 8.51e15iT - 1.48e32T^{2} \)
31 \( 1 + 6.89e15T + 6.45e32T^{2} \)
37 \( 1 + 1.87e17T + 3.16e34T^{2} \)
41 \( 1 + 3.13e16iT - 3.02e35T^{2} \)
43 \( 1 - 1.21e18T + 8.63e35T^{2} \)
47 \( 1 + 8.24e16iT - 6.11e36T^{2} \)
53 \( 1 + 1.29e19iT - 8.59e37T^{2} \)
59 \( 1 + 3.63e19iT - 9.09e38T^{2} \)
61 \( 1 - 1.41e19T + 1.89e39T^{2} \)
67 \( 1 + 1.54e20T + 1.49e40T^{2} \)
71 \( 1 - 3.03e20iT - 5.34e40T^{2} \)
73 \( 1 - 1.34e20T + 9.84e40T^{2} \)
79 \( 1 + 8.53e20T + 5.59e41T^{2} \)
83 \( 1 + 1.69e21iT - 1.65e42T^{2} \)
89 \( 1 - 4.69e21iT - 7.70e42T^{2} \)
97 \( 1 + 1.29e21T + 5.11e43T^{2} \)
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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

−17.34654960527169121242746973668, −15.91483730559461010507569854109, −14.58952992140480893943337352139, −13.46957929098123120296481208472, −10.97981619216522221360195293271, −8.746554557197146115141080645029, −8.067757055664218551949908863347, −5.40189581110705210435723816819, −3.90710419240821869368418732219, −1.42058027988970993848839809919, 1.36193409838823260870900480250, 2.53470554810822639501796941672, 4.47412004923420377580163612198, 7.29272195269838612357597750896, 8.782700711411725233199778644293, 10.71665074625163231701327714126, 12.28689167033715627534830702988, 14.04190391780233732624808646996, 14.93105011655221327891463838449, 17.87195806517341825903310469775

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