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Properties

Label 2-21-3.2-c6-0-6
Degree $2$
Conductor $21$
Sign $0.946 - 0.322i$
Analytic cond. $4.83113$
Root an. cond. $2.19798$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.92i·2-s + (25.5 − 8.70i)3-s + 28.9·4-s − 144. i·5-s + (51.5 + 151. i)6-s + 129.·7-s + 550. i·8-s + (577. − 444. i)9-s + 855.·10-s + 1.76e3i·11-s + (739. − 251. i)12-s − 1.25e3·13-s + 767. i·14-s + (−1.25e3 − 3.69e3i)15-s − 1.40e3·16-s − 6.02e3i·17-s + ⋯
L(s)  = 1  + 0.740i·2-s + (0.946 − 0.322i)3-s + 0.452·4-s − 1.15i·5-s + (0.238 + 0.700i)6-s + 0.377·7-s + 1.07i·8-s + (0.792 − 0.610i)9-s + 0.855·10-s + 1.32i·11-s + (0.427 − 0.145i)12-s − 0.573·13-s + 0.279i·14-s + (−0.372 − 1.09i)15-s − 0.343·16-s − 1.22i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.946 - 0.322i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.946 - 0.322i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $0.946 - 0.322i$
Analytic conductor: \(4.83113\)
Root analytic conductor: \(2.19798\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :3),\ 0.946 - 0.322i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(2.24393 + 0.371518i\)
\(L(\frac12)\) \(\approx\) \(2.24393 + 0.371518i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-25.5 + 8.70i)T \)
7 \( 1 - 129.T \)
good2 \( 1 - 5.92iT - 64T^{2} \)
5 \( 1 + 144. iT - 1.56e4T^{2} \)
11 \( 1 - 1.76e3iT - 1.77e6T^{2} \)
13 \( 1 + 1.25e3T + 4.82e6T^{2} \)
17 \( 1 + 6.02e3iT - 2.41e7T^{2} \)
19 \( 1 + 9.37e3T + 4.70e7T^{2} \)
23 \( 1 - 1.53e4iT - 1.48e8T^{2} \)
29 \( 1 + 1.70e4iT - 5.94e8T^{2} \)
31 \( 1 + 2.05e4T + 8.87e8T^{2} \)
37 \( 1 - 8.46e3T + 2.56e9T^{2} \)
41 \( 1 - 5.88e4iT - 4.75e9T^{2} \)
43 \( 1 + 1.35e5T + 6.32e9T^{2} \)
47 \( 1 - 2.80e4iT - 1.07e10T^{2} \)
53 \( 1 + 1.25e5iT - 2.21e10T^{2} \)
59 \( 1 - 1.81e5iT - 4.21e10T^{2} \)
61 \( 1 - 3.00e5T + 5.15e10T^{2} \)
67 \( 1 - 3.14e5T + 9.04e10T^{2} \)
71 \( 1 + 4.38e5iT - 1.28e11T^{2} \)
73 \( 1 - 4.98e4T + 1.51e11T^{2} \)
79 \( 1 - 5.05e5T + 2.43e11T^{2} \)
83 \( 1 - 3.41e5iT - 3.26e11T^{2} \)
89 \( 1 + 8.24e5iT - 4.96e11T^{2} \)
97 \( 1 + 1.42e5T + 8.32e11T^{2} \)
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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

−16.77735957073526067189957200742, −15.46652459673983027513878380585, −14.63583225512548085763458390227, −13.12128847604963124081423466386, −11.89962611398839106572156501994, −9.557865930376331784109652697680, −8.183115980589145913666096401924, −7.06817428276081111854303713528, −4.84489684295728465501697310484, −1.97951011247600285315682796959, 2.27323561465292267517968495733, 3.61037685467188472779699927690, 6.70242705185363674647549710029, 8.424911793720795758166843500633, 10.36952149858340173709901490911, 10.97587544056310368934095558846, 12.79566374692639346218659627319, 14.38473290091707562721724364220, 15.13735754230260255633504838684, 16.61883684232836490383690323608

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