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Properties

Label 2-6-3.2-c20-0-1
Degree $2$
Conductor $6$
Sign $0.213 - 0.977i$
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 + (−5.76e4 − 1.25e4i)3-s − 5.24e5·4-s − 8.94e6i·5-s + (9.10e6 − 4.17e7i)6-s − 3.40e7·7-s − 3.79e8i·8-s + (3.17e9 + 1.45e9i)9-s + 6.47e9·10-s + 6.27e9i·11-s + (3.02e10 + 6.59e9i)12-s − 1.12e11·13-s − 2.46e10i·14-s + (−1.12e11 + 5.16e11i)15-s + 2.74e11·16-s + 3.26e12i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.977 − 0.213i)3-s − 0.500·4-s − 0.916i·5-s + (0.150 − 0.690i)6-s − 0.120·7-s − 0.353i·8-s + (0.909 + 0.416i)9-s + 0.647·10-s + 0.241i·11-s + (0.488 + 0.106i)12-s − 0.818·13-s − 0.0851i·14-s + (−0.195 + 0.895i)15-s + 0.250·16-s + 1.62i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.213 - 0.977i)\, \overline{\Lambda}(21-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s+10) \, L(s)\cr =\mathstrut & (0.213 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6\)    =    \(2 \cdot 3\)
Sign: $0.213 - 0.977i$
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: \((2,\ 6,\ (\ :10),\ 0.213 - 0.977i)\)

Particular Values

\(L(\frac{21}{2})\) \(\approx\) \(0.790409 + 0.636624i\)
\(L(\frac12)\) \(\approx\) \(0.790409 + 0.636624i\)
\(L(11)\) 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 \( 1 + (5.76e4 + 1.25e4i)T \)
good5 \( 1 + 8.94e6iT - 9.53e13T^{2} \)
7 \( 1 + 3.40e7T + 7.97e16T^{2} \)
11 \( 1 - 6.27e9iT - 6.72e20T^{2} \)
13 \( 1 + 1.12e11T + 1.90e22T^{2} \)
17 \( 1 - 3.26e12iT - 4.06e24T^{2} \)
19 \( 1 - 8.13e12T + 3.75e25T^{2} \)
23 \( 1 - 5.77e13iT - 1.71e27T^{2} \)
29 \( 1 + 6.61e14iT - 1.76e29T^{2} \)
31 \( 1 - 1.33e15T + 6.71e29T^{2} \)
37 \( 1 - 1.65e15T + 2.31e31T^{2} \)
41 \( 1 - 1.90e16iT - 1.80e32T^{2} \)
43 \( 1 - 1.96e16T + 4.67e32T^{2} \)
47 \( 1 + 2.59e16iT - 2.76e33T^{2} \)
53 \( 1 - 1.97e17iT - 3.05e34T^{2} \)
59 \( 1 - 7.73e17iT - 2.61e35T^{2} \)
61 \( 1 + 1.23e17T + 5.08e35T^{2} \)
67 \( 1 + 3.35e17T + 3.32e36T^{2} \)
71 \( 1 + 1.61e18iT - 1.05e37T^{2} \)
73 \( 1 - 3.68e17T + 1.84e37T^{2} \)
79 \( 1 - 2.03e18T + 8.96e37T^{2} \)
83 \( 1 + 8.34e18iT - 2.40e38T^{2} \)
89 \( 1 + 1.06e19iT - 9.72e38T^{2} \)
97 \( 1 + 6.98e19T + 5.43e39T^{2} \)
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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

−17.64544268975850349707742443724, −16.79770862865476391078764057828, −15.45668175297868211682928888587, −13.27979326576849070123862980038, −11.98876114115800650687230835951, −9.791133441841608671816520051088, −7.74423773221582254248572556015, −5.94620326678040710867585477599, −4.58533525425514943290684505388, −1.09625541028191109385118157623, 0.58853202300505844344429604279, 2.91552539540165868070113004409, 4.97202153329955050031527367975, 6.94786738621814409515808933663, 9.747642032819242865489301738118, 11.02055535536449110887945796098, 12.24645361960974966955866827675, 14.23084565160043941512318736277, 16.13745896219841920126176674883, 17.83793005525955425343441416444

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