Properties

Label 2-21-7.3-c6-0-0
Degree $2$
Conductor $21$
Sign $0.724 - 0.689i$
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  + (−7.79 − 13.5i)2-s + (−13.5 − 7.79i)3-s + (−89.6 + 155. i)4-s + (22.3 − 12.9i)5-s + 243. i·6-s + (−203. − 276. i)7-s + 1.79e3·8-s + (121.5 + 210. i)9-s + (−348. − 201. i)10-s + (−311. + 540. i)11-s + (2.41e3 − 1.39e3i)12-s + 3.25e3i·13-s + (−2.14e3 + 4.89e3i)14-s − 402.·15-s + (−8.27e3 − 1.43e4i)16-s + (275. + 158. i)17-s + ⋯
L(s)  = 1  + (−0.974 − 1.68i)2-s + (−0.5 − 0.288i)3-s + (−1.40 + 2.42i)4-s + (0.178 − 0.103i)5-s + 1.12i·6-s + (−0.592 − 0.805i)7-s + 3.50·8-s + (0.166 + 0.288i)9-s + (−0.348 − 0.201i)10-s + (−0.234 + 0.405i)11-s + (1.40 − 0.808i)12-s + 1.48i·13-s + (−0.783 + 1.78i)14-s − 0.119·15-s + (−2.02 − 3.49i)16-s + (0.0560 + 0.0323i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.724 - 0.689i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.724 - 0.689i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $0.724 - 0.689i$
Analytic conductor: \(4.83113\)
Root analytic conductor: \(2.19798\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :3),\ 0.724 - 0.689i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.126548 + 0.0505833i\)
\(L(\frac12)\) \(\approx\) \(0.126548 + 0.0505833i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (13.5 + 7.79i)T \)
7 \( 1 + (203. + 276. i)T \)
good2 \( 1 + (7.79 + 13.5i)T + (-32 + 55.4i)T^{2} \)
5 \( 1 + (-22.3 + 12.9i)T + (7.81e3 - 1.35e4i)T^{2} \)
11 \( 1 + (311. - 540. i)T + (-8.85e5 - 1.53e6i)T^{2} \)
13 \( 1 - 3.25e3iT - 4.82e6T^{2} \)
17 \( 1 + (-275. - 158. i)T + (1.20e7 + 2.09e7i)T^{2} \)
19 \( 1 + (5.19e3 - 3.00e3i)T + (2.35e7 - 4.07e7i)T^{2} \)
23 \( 1 + (-21.0 - 36.3i)T + (-7.40e7 + 1.28e8i)T^{2} \)
29 \( 1 + 2.42e4T + 5.94e8T^{2} \)
31 \( 1 + (1.75e4 + 1.01e4i)T + (4.43e8 + 7.68e8i)T^{2} \)
37 \( 1 + (-8.68e3 - 1.50e4i)T + (-1.28e9 + 2.22e9i)T^{2} \)
41 \( 1 + 1.00e5iT - 4.75e9T^{2} \)
43 \( 1 + 6.78e4T + 6.32e9T^{2} \)
47 \( 1 + (-5.70e4 + 3.29e4i)T + (5.38e9 - 9.33e9i)T^{2} \)
53 \( 1 + (4.96e4 - 8.59e4i)T + (-1.10e10 - 1.91e10i)T^{2} \)
59 \( 1 + (-8.71e4 - 5.03e4i)T + (2.10e10 + 3.65e10i)T^{2} \)
61 \( 1 + (2.01e5 - 1.16e5i)T + (2.57e10 - 4.46e10i)T^{2} \)
67 \( 1 + (2.05e4 - 3.55e4i)T + (-4.52e10 - 7.83e10i)T^{2} \)
71 \( 1 + 4.00e5T + 1.28e11T^{2} \)
73 \( 1 + (-4.79e5 - 2.76e5i)T + (7.56e10 + 1.31e11i)T^{2} \)
79 \( 1 + (-5.24e3 - 9.07e3i)T + (-1.21e11 + 2.10e11i)T^{2} \)
83 \( 1 + 7.12e5iT - 3.26e11T^{2} \)
89 \( 1 + (1.64e5 - 9.52e4i)T + (2.48e11 - 4.30e11i)T^{2} \)
97 \( 1 + 1.07e6iT - 8.32e11T^{2} \)
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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

−17.19538794618818688979929812888, −16.62546626812571063873788134537, −13.63650055281389018728876374100, −12.64498884056846365063359182765, −11.44831060437263260574094191385, −10.31129830086727870455778204843, −9.174400056937278920682653838985, −7.32789561451154736016953824340, −4.02850297875057390650102706519, −1.76558587244550522737984977200, 0.12528835614005209793905094144, 5.36806603039245769126596831090, 6.36655293360383032363135203511, 8.102330387234781938401250412642, 9.447674631426078148180535429733, 10.62370957658713438050822607245, 13.10574733309301088339925794123, 14.84585379154219749065090676080, 15.65031816761995705455329359771, 16.59663357250783592602179475317

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