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Properties

Label 2-6-3.2-c22-0-0
Degree $2$
Conductor $6$
Sign $-0.981 + 0.191i$
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 + (3.39e4 + 1.73e5i)3-s − 2.09e6·4-s + 7.14e7i·5-s + (2.51e8 − 4.91e7i)6-s − 1.07e9·7-s + 3.03e9i·8-s + (−2.90e10 + 1.18e10i)9-s + 1.03e11·10-s − 5.54e11i·11-s + (−7.12e10 − 3.64e11i)12-s + 7.40e11·13-s + 1.55e12i·14-s + (−1.24e13 + 2.42e12i)15-s + 4.39e12·16-s − 3.57e12i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.191 + 0.981i)3-s − 0.500·4-s + 1.46i·5-s + (0.693 − 0.135i)6-s − 0.544·7-s + 0.353i·8-s + (−0.926 + 0.376i)9-s + 1.03·10-s − 1.94i·11-s + (−0.0958 − 0.490i)12-s + 0.413·13-s + 0.385i·14-s + (−1.43 + 0.280i)15-s + 0.250·16-s − 0.104i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 + 0.191i)\, \overline{\Lambda}(23-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s+11) \, L(s)\cr =\mathstrut & (-0.981 + 0.191i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6\)    =    \(2 \cdot 3\)
Sign: $-0.981 + 0.191i$
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: \((2,\ 6,\ (\ :11),\ -0.981 + 0.191i)\)

Particular Values

\(L(\frac{23}{2})\) \(\approx\) \(0.0215550 - 0.222825i\)
\(L(\frac12)\) \(\approx\) \(0.0215550 - 0.222825i\)
\(L(12)\) 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 \( 1 + (-3.39e4 - 1.73e5i)T \)
good5 \( 1 - 7.14e7iT - 2.38e15T^{2} \)
7 \( 1 + 1.07e9T + 3.90e18T^{2} \)
11 \( 1 + 5.54e11iT - 8.14e22T^{2} \)
13 \( 1 - 7.40e11T + 3.21e24T^{2} \)
17 \( 1 + 3.57e12iT - 1.17e27T^{2} \)
19 \( 1 + 2.06e14T + 1.35e28T^{2} \)
23 \( 1 - 4.26e13iT - 9.07e29T^{2} \)
29 \( 1 - 8.91e15iT - 1.48e32T^{2} \)
31 \( 1 + 1.01e16T + 6.45e32T^{2} \)
37 \( 1 - 1.51e17T + 3.16e34T^{2} \)
41 \( 1 + 2.66e17iT - 3.02e35T^{2} \)
43 \( 1 + 6.89e17T + 8.63e35T^{2} \)
47 \( 1 + 3.30e18iT - 6.11e36T^{2} \)
53 \( 1 - 8.58e18iT - 8.59e37T^{2} \)
59 \( 1 - 3.62e19iT - 9.09e38T^{2} \)
61 \( 1 + 5.79e19T + 1.89e39T^{2} \)
67 \( 1 + 1.39e20T + 1.49e40T^{2} \)
71 \( 1 - 3.40e20iT - 5.34e40T^{2} \)
73 \( 1 + 1.67e19T + 9.84e40T^{2} \)
79 \( 1 - 6.94e20T + 5.59e41T^{2} \)
83 \( 1 - 2.26e20iT - 1.65e42T^{2} \)
89 \( 1 + 7.87e20iT - 7.70e42T^{2} \)
97 \( 1 + 4.63e21T + 5.11e43T^{2} \)
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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

−18.59046298409968519308760322388, −16.55018926498111031085542125542, −14.91443381448105831316177763202, −13.66283302484622756158467021297, −11.15794279573643159791542321306, −10.46253415353980878289340349988, −8.704985582554747927158710356599, −6.09495039423803243113102479723, −3.69332664456683175345029923095, −2.78652067886083854168466816341, 0.07868669097901799890719335962, 1.73250506297629130202421026732, 4.55436833473363802863138654614, 6.39435692315290611878426193626, 7.995862286365082450215271174170, 9.342757364773974348884695953177, 12.47805967612726875844789984513, 13.09085188127538681467872867447, 15.03413017567110749409173287062, 16.72968777875830010510305790258

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