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Properties

Label 2-6-3.2-c20-0-4
Degree $2$
Conductor $6$
Sign $0.903 + 0.428i$
Analytic cond. $15.2108$
Root an. cond. $3.90010$
Motivic weight $20$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 724. i·2-s + (2.53e4 − 5.33e4i)3-s − 5.24e5·4-s + 6.05e6i·5-s + (3.86e7 + 1.83e7i)6-s + 1.14e8·7-s − 3.79e8i·8-s + (−2.20e9 − 2.70e9i)9-s − 4.38e9·10-s − 2.84e10i·11-s + (−1.32e10 + 2.79e10i)12-s + 1.97e11·13-s + 8.27e10i·14-s + (3.22e11 + 1.53e11i)15-s + 2.74e11·16-s − 2.02e12i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.428 − 0.903i)3-s − 0.500·4-s + 0.619i·5-s + (0.638 + 0.303i)6-s + 0.404·7-s − 0.353i·8-s + (−0.632 − 0.774i)9-s − 0.438·10-s − 1.09i·11-s + (−0.214 + 0.451i)12-s + 1.43·13-s + 0.285i·14-s + (0.559 + 0.265i)15-s + 0.250·16-s − 1.00i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.903 + 0.428i)\, \overline{\Lambda}(21-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s+10) \, L(s)\cr =\mathstrut & (0.903 + 0.428i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6\)    =    \(2 \cdot 3\)
Sign: $0.903 + 0.428i$
Analytic conductor: \(15.2108\)
Root analytic conductor: \(3.90010\)
Motivic weight: \(20\)
Rational: no
Arithmetic: yes
Character: $\chi_{6} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 6,\ (\ :10),\ 0.903 + 0.428i)\)

Particular Values

\(L(\frac{21}{2})\) \(\approx\) \(2.03394 - 0.458281i\)
\(L(\frac12)\) \(\approx\) \(2.03394 - 0.458281i\)
\(L(11)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 724. iT \)
3 \( 1 + (-2.53e4 + 5.33e4i)T \)
good5 \( 1 - 6.05e6iT - 9.53e13T^{2} \)
7 \( 1 - 1.14e8T + 7.97e16T^{2} \)
11 \( 1 + 2.84e10iT - 6.72e20T^{2} \)
13 \( 1 - 1.97e11T + 1.90e22T^{2} \)
17 \( 1 + 2.02e12iT - 4.06e24T^{2} \)
19 \( 1 - 9.02e12T + 3.75e25T^{2} \)
23 \( 1 + 1.39e13iT - 1.71e27T^{2} \)
29 \( 1 + 1.81e14iT - 1.76e29T^{2} \)
31 \( 1 + 9.52e14T + 6.71e29T^{2} \)
37 \( 1 - 4.96e15T + 2.31e31T^{2} \)
41 \( 1 + 2.06e16iT - 1.80e32T^{2} \)
43 \( 1 + 2.15e16T + 4.67e32T^{2} \)
47 \( 1 - 7.74e16iT - 2.76e33T^{2} \)
53 \( 1 - 1.50e17iT - 3.05e34T^{2} \)
59 \( 1 - 7.95e17iT - 2.61e35T^{2} \)
61 \( 1 - 1.04e18T + 5.08e35T^{2} \)
67 \( 1 + 5.49e17T + 3.32e36T^{2} \)
71 \( 1 - 3.52e16iT - 1.05e37T^{2} \)
73 \( 1 + 5.86e18T + 1.84e37T^{2} \)
79 \( 1 - 5.32e18T + 8.96e37T^{2} \)
83 \( 1 + 2.31e19iT - 2.40e38T^{2} \)
89 \( 1 + 2.31e19iT - 9.72e38T^{2} \)
97 \( 1 - 2.76e19T + 5.43e39T^{2} \)
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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

−18.07149819139386982975290835470, −16.12925433727724997011684110086, −14.38232446466558865278229889555, −13.46662246451912207549947858719, −11.35382576658432525591152387622, −8.822231336810565096669404867069, −7.40551908163622277228948329436, −5.92432762826114702856633351306, −3.19715540455368673198155432279, −0.925983723823246647355197967390, 1.54250502388213930850023374692, 3.61721825048883441673806316719, 5.06432788519136776259195733609, 8.355488090684715689356521127908, 9.728748564268908379709136965197, 11.26737936233567643605808482584, 13.12773438687123182052659912510, 14.77984916152432091653340246512, 16.36182569060859908284995582357, 18.06986008400235109978444061874

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