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Properties

Label 2-6-3.2-c10-0-3
Degree $2$
Conductor $6$
Sign $-0.566 + 0.824i$
Analytic cond. $3.81214$
Root an. cond. $1.95247$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 22.6i·2-s + (−200. − 137. i)3-s − 512.·4-s − 3.63e3i·5-s + (3.11e3 − 4.53e3i)6-s − 2.32e4·7-s − 1.15e4i·8-s + (2.11e4 + 5.51e4i)9-s + 8.21e4·10-s + 6.24e4i·11-s + (1.02e5 + 7.04e4i)12-s − 1.70e5·13-s − 5.25e5i·14-s + (−4.99e5 + 7.27e5i)15-s + 2.62e5·16-s − 2.66e6i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.824 − 0.566i)3-s − 0.500·4-s − 1.16i·5-s + (0.400 − 0.582i)6-s − 1.38·7-s − 0.353i·8-s + (0.358 + 0.933i)9-s + 0.821·10-s + 0.387i·11-s + (0.412 + 0.283i)12-s − 0.458·13-s − 0.977i·14-s + (−0.657 + 0.957i)15-s + 0.250·16-s − 1.87i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.566 + 0.824i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.566 + 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6\)    =    \(2 \cdot 3\)
Sign: $-0.566 + 0.824i$
Analytic conductor: \(3.81214\)
Root analytic conductor: \(1.95247\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{6} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 6,\ (\ :5),\ -0.566 + 0.824i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.201661 - 0.383272i\)
\(L(\frac12)\) \(\approx\) \(0.201661 - 0.383272i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 22.6iT \)
3 \( 1 + (200. + 137. i)T \)
good5 \( 1 + 3.63e3iT - 9.76e6T^{2} \)
7 \( 1 + 2.32e4T + 2.82e8T^{2} \)
11 \( 1 - 6.24e4iT - 2.59e10T^{2} \)
13 \( 1 + 1.70e5T + 1.37e11T^{2} \)
17 \( 1 + 2.66e6iT - 2.01e12T^{2} \)
19 \( 1 - 7.66e5T + 6.13e12T^{2} \)
23 \( 1 - 1.40e6iT - 4.14e13T^{2} \)
29 \( 1 - 4.83e6iT - 4.20e14T^{2} \)
31 \( 1 + 4.18e7T + 8.19e14T^{2} \)
37 \( 1 - 5.01e7T + 4.80e15T^{2} \)
41 \( 1 + 1.49e8iT - 1.34e16T^{2} \)
43 \( 1 + 1.98e8T + 2.16e16T^{2} \)
47 \( 1 + 1.55e8iT - 5.25e16T^{2} \)
53 \( 1 - 4.21e7iT - 1.74e17T^{2} \)
59 \( 1 - 2.92e8iT - 5.11e17T^{2} \)
61 \( 1 + 5.30e8T + 7.13e17T^{2} \)
67 \( 1 - 5.22e8T + 1.82e18T^{2} \)
71 \( 1 - 5.71e8iT - 3.25e18T^{2} \)
73 \( 1 - 2.18e9T + 4.29e18T^{2} \)
79 \( 1 - 1.96e9T + 9.46e18T^{2} \)
83 \( 1 + 2.18e9iT - 1.55e19T^{2} \)
89 \( 1 + 2.38e8iT - 3.11e19T^{2} \)
97 \( 1 + 8.84e9T + 7.37e19T^{2} \)
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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

−20.00548208426430892695015349209, −18.36785679444259821864030742450, −16.77597424596093827689795094809, −16.03763892681652958420284373008, −13.37281674208131396877229763228, −12.26328058055213093332307633811, −9.466190169821971645186608656230, −7.09930741215260245259768013519, −5.21672670865504311748330196993, −0.32804557147860223561523458314, 3.44872814696230968742953399596, 6.30536887493922036140559292646, 9.831638671473870957217108698789, 10.96069053696150353265466271595, 12.73329651514715162216073981973, 14.94666768885243624353242931732, 16.69932341186755499683288247564, 18.37726049619460841559478098667, 19.62239729369285249979443177001, 21.71936361150500820581503208694

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