Properties

Label 2-6-3.2-c26-0-4
Degree $2$
Conductor $6$
Sign $0.969 + 0.243i$
Analytic cond. $25.6975$
Root an. cond. $5.06927$
Motivic weight $26$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.79e3i·2-s + (−3.88e5 + 1.54e6i)3-s − 3.35e7·4-s + 1.99e8i·5-s + (−8.95e9 − 2.25e9i)6-s − 4.03e9·7-s − 1.94e11i·8-s + (−2.23e12 − 1.20e12i)9-s − 1.15e12·10-s − 4.12e13i·11-s + (1.30e13 − 5.18e13i)12-s − 9.55e13·13-s − 2.33e13i·14-s + (−3.08e14 − 7.74e13i)15-s + 1.12e15·16-s − 5.48e15i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.243 + 0.969i)3-s − 0.500·4-s + 0.163i·5-s + (−0.685 − 0.172i)6-s − 0.0416·7-s − 0.353i·8-s + (−0.881 − 0.472i)9-s − 0.115·10-s − 1.19i·11-s + (0.121 − 0.484i)12-s − 0.315·13-s − 0.0294i·14-s + (−0.158 − 0.0397i)15-s + 0.250·16-s − 0.553i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 + 0.243i)\, \overline{\Lambda}(27-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s+13) \, L(s)\cr =\mathstrut & (0.969 + 0.243i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6\)    =    \(2 \cdot 3\)
Sign: $0.969 + 0.243i$
Analytic conductor: \(25.6975\)
Root analytic conductor: \(5.06927\)
Motivic weight: \(26\)
Rational: no
Arithmetic: yes
Character: $\chi_{6} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 6,\ (\ :13),\ 0.969 + 0.243i)\)

Particular Values

\(L(\frac{27}{2})\) \(\approx\) \(0.8897940130\)
\(L(\frac12)\) \(\approx\) \(0.8897940130\)
\(L(14)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 5.79e3iT \)
3 \( 1 + (3.88e5 - 1.54e6i)T \)
good5 \( 1 - 1.99e8iT - 1.49e18T^{2} \)
7 \( 1 + 4.03e9T + 9.38e21T^{2} \)
11 \( 1 + 4.12e13iT - 1.19e27T^{2} \)
13 \( 1 + 9.55e13T + 9.17e28T^{2} \)
17 \( 1 + 5.48e15iT - 9.81e31T^{2} \)
19 \( 1 + 2.57e16T + 1.76e33T^{2} \)
23 \( 1 - 6.39e17iT - 2.54e35T^{2} \)
29 \( 1 + 1.27e19iT - 1.05e38T^{2} \)
31 \( 1 - 4.74e19T + 5.96e38T^{2} \)
37 \( 1 + 6.19e19T + 5.93e40T^{2} \)
41 \( 1 - 1.36e20iT - 8.55e41T^{2} \)
43 \( 1 + 2.89e21T + 2.95e42T^{2} \)
47 \( 1 + 1.44e21iT - 2.98e43T^{2} \)
53 \( 1 + 1.68e22iT - 6.77e44T^{2} \)
59 \( 1 + 1.69e23iT - 1.10e46T^{2} \)
61 \( 1 - 1.07e23T + 2.62e46T^{2} \)
67 \( 1 - 7.95e22T + 3.00e47T^{2} \)
71 \( 1 + 1.77e24iT - 1.35e48T^{2} \)
73 \( 1 + 2.59e24T + 2.79e48T^{2} \)
79 \( 1 - 5.88e24T + 2.17e49T^{2} \)
83 \( 1 + 7.37e24iT - 7.87e49T^{2} \)
89 \( 1 + 7.84e24iT - 4.83e50T^{2} \)
97 \( 1 + 7.65e25T + 4.52e51T^{2} \)
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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

−16.28828805693883957634362993589, −15.13782598281303317168924074413, −13.73054829117253263735663142511, −11.47767152019588099177340752604, −9.874358103065890881415306037885, −8.386001498549118927465418549093, −6.30888970840638005891640851077, −4.89439105405847051435505365445, −3.26333835483676900168983272453, −0.33471400306106074123376869757, 1.24426973634046856035387547989, 2.55424317825278415833738360212, 4.75697089245883205568186985839, 6.73197568481197597334589843825, 8.469473921179877256023160356645, 10.40523319793172939067129621141, 12.09740905553771183780408249953, 12.96073034760678440321631541977, 14.62587839464113786103846499896, 16.98498161212748471636600994471

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