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Properties

Label 2-21-21.20-c21-0-15
Degree $2$
Conductor $21$
Sign $0.456 + 0.889i$
Analytic cond. $58.6902$
Root an. cond. $7.66095$
Motivic weight $21$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.06e3i·2-s + (759. + 1.02e5i)3-s − 2.15e6·4-s − 1.95e7·5-s + (2.10e8 − 1.56e6i)6-s + (−6.67e8 + 3.36e8i)7-s + 1.15e8i·8-s + (−1.04e10 + 1.55e8i)9-s + 4.02e10i·10-s − 1.09e11i·11-s + (−1.63e9 − 2.20e11i)12-s + 5.15e11i·13-s + (6.93e11 + 1.37e12i)14-s + (−1.48e10 − 1.99e12i)15-s − 4.27e12·16-s − 1.12e13·17-s + ⋯
L(s)  = 1  − 1.42i·2-s + (0.00742 + 0.999i)3-s − 1.02·4-s − 0.894·5-s + (1.42 − 0.0105i)6-s + (−0.892 + 0.450i)7-s + 0.0381i·8-s + (−0.999 + 0.0148i)9-s + 1.27i·10-s − 1.27i·11-s + (−0.00762 − 1.02i)12-s + 1.03i·13-s + (0.640 + 1.27i)14-s + (−0.00663 − 0.893i)15-s − 0.972·16-s − 1.35·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.456 + 0.889i)\, \overline{\Lambda}(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & (0.456 + 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $0.456 + 0.889i$
Analytic conductor: \(58.6902\)
Root analytic conductor: \(7.66095\)
Motivic weight: \(21\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :21/2),\ 0.456 + 0.889i)\)

Particular Values

\(L(11)\) \(\approx\) \(0.7853473954\)
\(L(\frac12)\) \(\approx\) \(0.7853473954\)
\(L(\frac{23}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-759. - 1.02e5i)T \)
7 \( 1 + (6.67e8 - 3.36e8i)T \)
good2 \( 1 + 2.06e3iT - 2.09e6T^{2} \)
5 \( 1 + 1.95e7T + 4.76e14T^{2} \)
11 \( 1 + 1.09e11iT - 7.40e21T^{2} \)
13 \( 1 - 5.15e11iT - 2.47e23T^{2} \)
17 \( 1 + 1.12e13T + 6.90e25T^{2} \)
19 \( 1 - 4.21e13iT - 7.14e26T^{2} \)
23 \( 1 + 2.30e14iT - 3.94e28T^{2} \)
29 \( 1 - 2.95e15iT - 5.13e30T^{2} \)
31 \( 1 + 4.09e15iT - 2.08e31T^{2} \)
37 \( 1 - 2.53e16T + 8.55e32T^{2} \)
41 \( 1 - 1.05e17T + 7.38e33T^{2} \)
43 \( 1 - 2.13e17T + 2.00e34T^{2} \)
47 \( 1 + 2.13e17T + 1.30e35T^{2} \)
53 \( 1 + 1.61e18iT - 1.62e36T^{2} \)
59 \( 1 + 3.38e18T + 1.54e37T^{2} \)
61 \( 1 + 5.43e18iT - 3.10e37T^{2} \)
67 \( 1 - 1.42e19T + 2.22e38T^{2} \)
71 \( 1 - 2.98e19iT - 7.52e38T^{2} \)
73 \( 1 - 3.08e18iT - 1.34e39T^{2} \)
79 \( 1 + 6.07e19T + 7.08e39T^{2} \)
83 \( 1 + 3.37e19T + 1.99e40T^{2} \)
89 \( 1 - 4.45e18T + 8.65e40T^{2} \)
97 \( 1 - 8.29e20iT - 5.27e41T^{2} \)
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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

−12.72887177489284078742546381751, −11.55367908907069895839084206802, −10.86346095745128198171283059262, −9.543405067216272717924055249374, −8.584183637734736424250819692357, −6.22042891193340899318143485703, −4.24672582377538427939104364520, −3.54576381566803816648352910052, −2.40717049061731187666132240838, −0.42988774397693910087749316871, 0.48377175590763188246941945610, 2.58382148487609523864043530755, 4.46412454079096699719314908800, 6.07921287027793424925012460660, 7.19678978067192649984110735453, 7.66800432438616680126260903775, 9.160119810094878301692860433522, 11.26036218990660492161219233745, 12.75166254708372688860461832348, 13.65065514130324137403411298764

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