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Properties

Label 2-6-3.2-c12-0-0
Degree $2$
Conductor $6$
Sign $-0.999 + 0.00605i$
Analytic cond. $5.48396$
Root an. cond. $2.34178$
Motivic weight $12$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 45.2i·2-s + (4.41 + 728. i)3-s − 2.04e3·4-s − 1.79e3i·5-s + (−3.29e4 + 199. i)6-s − 1.36e5·7-s − 9.26e4i·8-s + (−5.31e5 + 6.43e3i)9-s + 8.11e4·10-s + 1.76e6i·11-s + (−9.03e3 − 1.49e6i)12-s + 6.95e6·13-s − 6.18e6i·14-s + (1.30e6 − 7.91e3i)15-s + 4.19e6·16-s + 3.90e7i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.00605 + 0.999i)3-s − 0.500·4-s − 0.114i·5-s + (−0.707 + 0.00427i)6-s − 1.16·7-s − 0.353i·8-s + (−0.999 + 0.0121i)9-s + 0.0811·10-s + 0.993i·11-s + (−0.00302 − 0.499i)12-s + 1.44·13-s − 0.821i·14-s + (0.114 − 0.000694i)15-s + 0.250·16-s + 1.61i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.00605i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (-0.999 + 0.00605i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6\)    =    \(2 \cdot 3\)
Sign: $-0.999 + 0.00605i$
Analytic conductor: \(5.48396\)
Root analytic conductor: \(2.34178\)
Motivic weight: \(12\)
Rational: no
Arithmetic: yes
Character: $\chi_{6} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 6,\ (\ :6),\ -0.999 + 0.00605i)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(0.00300231 - 0.992271i\)
\(L(\frac12)\) \(\approx\) \(0.00300231 - 0.992271i\)
\(L(7)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 45.2iT \)
3 \( 1 + (-4.41 - 728. i)T \)
good5 \( 1 + 1.79e3iT - 2.44e8T^{2} \)
7 \( 1 + 1.36e5T + 1.38e10T^{2} \)
11 \( 1 - 1.76e6iT - 3.13e12T^{2} \)
13 \( 1 - 6.95e6T + 2.32e13T^{2} \)
17 \( 1 - 3.90e7iT - 5.82e14T^{2} \)
19 \( 1 + 6.02e7T + 2.21e15T^{2} \)
23 \( 1 - 1.16e8iT - 2.19e16T^{2} \)
29 \( 1 + 1.81e8iT - 3.53e17T^{2} \)
31 \( 1 - 2.39e8T + 7.87e17T^{2} \)
37 \( 1 + 1.10e8T + 6.58e18T^{2} \)
41 \( 1 + 3.21e9iT - 2.25e19T^{2} \)
43 \( 1 - 6.08e9T + 3.99e19T^{2} \)
47 \( 1 - 3.06e9iT - 1.16e20T^{2} \)
53 \( 1 + 2.76e10iT - 4.91e20T^{2} \)
59 \( 1 - 7.62e10iT - 1.77e21T^{2} \)
61 \( 1 - 3.69e10T + 2.65e21T^{2} \)
67 \( 1 + 5.88e10T + 8.18e21T^{2} \)
71 \( 1 - 1.03e11iT - 1.64e22T^{2} \)
73 \( 1 + 1.32e11T + 2.29e22T^{2} \)
79 \( 1 - 1.57e11T + 5.90e22T^{2} \)
83 \( 1 + 3.02e11iT - 1.06e23T^{2} \)
89 \( 1 + 5.40e10iT - 2.46e23T^{2} \)
97 \( 1 + 1.33e11T + 6.93e23T^{2} \)
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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

−21.03589885794088482287524330783, −19.37307585631186529512361477512, −17.32440871203312106465694628707, −16.04318673393528944742308922373, −14.97545168242198303995359744912, −12.98220827451204007180900712829, −10.37058294287042598683520438129, −8.787728935165934326639790483133, −6.16691504109089348090783432642, −3.94161750708168357409220545920, 0.60671937747199961067823339247, 2.98783260521782573627115847554, 6.37300899312216697516961753252, 8.774278129596321957654829837643, 11.05463723265938482434783667408, 12.75180564268946181586688757785, 13.84819788717521938121398068858, 16.40246511746883477196878208251, 18.38068027590648892330590943407, 19.15130035503680744109969518438

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