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Properties

Label 2-21-21.20-c29-0-58
Degree $2$
Conductor $21$
Sign $-0.643 - 0.765i$
Analytic cond. $111.883$
Root an. cond. $10.5775$
Motivic weight $29$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.18e4i·2-s + (−3.41e6 + 7.54e6i)3-s + 5.73e7·4-s − 2.19e10·5-s + (1.65e11 + 7.47e10i)6-s + (7.76e11 − 1.61e12i)7-s − 1.30e13i·8-s + (−4.53e13 − 5.15e13i)9-s + 4.81e14i·10-s − 2.40e15i·11-s + (−1.95e14 + 4.33e14i)12-s + 1.13e16i·13-s + (−3.54e16 − 1.70e16i)14-s + (7.49e16 − 1.65e17i)15-s − 2.54e17·16-s − 2.64e17·17-s + ⋯
L(s)  = 1  − 0.945i·2-s + (−0.412 + 0.911i)3-s + 0.106·4-s − 1.60·5-s + (0.861 + 0.389i)6-s + (0.432 − 0.901i)7-s − 1.04i·8-s + (−0.660 − 0.750i)9-s + 1.52i·10-s − 1.90i·11-s + (−0.0440 + 0.0973i)12-s + 0.798i·13-s + (−0.851 − 0.409i)14-s + (0.663 − 1.46i)15-s − 0.881·16-s − 0.380·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.643 - 0.765i)\, \overline{\Lambda}(30-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+29/2) \, L(s)\cr =\mathstrut & (-0.643 - 0.765i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $-0.643 - 0.765i$
Analytic conductor: \(111.883\)
Root analytic conductor: \(10.5775\)
Motivic weight: \(29\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :29/2),\ -0.643 - 0.765i)\)

Particular Values

\(L(15)\) \(\approx\) \(1.062035261\)
\(L(\frac12)\) \(\approx\) \(1.062035261\)
\(L(\frac{31}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (3.41e6 - 7.54e6i)T \)
7 \( 1 + (-7.76e11 + 1.61e12i)T \)
good2 \( 1 + 2.18e4iT - 5.36e8T^{2} \)
5 \( 1 + 2.19e10T + 1.86e20T^{2} \)
11 \( 1 + 2.40e15iT - 1.58e30T^{2} \)
13 \( 1 - 1.13e16iT - 2.01e32T^{2} \)
17 \( 1 + 2.64e17T + 4.81e35T^{2} \)
19 \( 1 + 3.62e18iT - 1.21e37T^{2} \)
23 \( 1 + 1.89e19iT - 3.09e39T^{2} \)
29 \( 1 + 1.75e21iT - 2.56e42T^{2} \)
31 \( 1 + 5.25e21iT - 1.77e43T^{2} \)
37 \( 1 - 7.21e22T + 3.00e45T^{2} \)
41 \( 1 + 6.20e22T + 5.89e46T^{2} \)
43 \( 1 - 4.32e23T + 2.34e47T^{2} \)
47 \( 1 - 2.59e24T + 3.09e48T^{2} \)
53 \( 1 - 2.18e24iT - 1.00e50T^{2} \)
59 \( 1 + 3.87e25T + 2.26e51T^{2} \)
61 \( 1 + 1.06e26iT - 5.95e51T^{2} \)
67 \( 1 + 3.98e26T + 9.04e52T^{2} \)
71 \( 1 + 9.94e26iT - 4.85e53T^{2} \)
73 \( 1 + 8.49e25iT - 1.08e54T^{2} \)
79 \( 1 - 1.57e27T + 1.07e55T^{2} \)
83 \( 1 - 3.67e27T + 4.50e55T^{2} \)
89 \( 1 + 2.12e28T + 3.40e56T^{2} \)
97 \( 1 - 7.01e28iT - 4.13e57T^{2} \)
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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

−11.15490325795278826001907786350, −10.87388560034267122546367603139, −9.160128290832813244643120141990, −7.80142713578969830733700477907, −6.35693660287060851709262978594, −4.37020061235193429709289156563, −3.89047580762375247252681873255, −2.82990501239637877751400836536, −0.69384306907578320413950876521, −0.36889789664298802497107282999, 1.45166171553615345644147198273, 2.66861492549741645498489329511, 4.59243482904731883644137672700, 5.66764889668664218399647326055, 7.09558677224972657289816763067, 7.63183031517376103349334055747, 8.529604424831893145820920015479, 10.87233791303398519129259360271, 12.04746338503934829077237873608, 12.46218095593280478886545215769

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