Properties

Label 2-21-21.2-c6-0-6
Degree $2$
Conductor $21$
Sign $0.0670 - 0.997i$
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  + (7.42 + 4.28i)2-s + (6.70 + 26.1i)3-s + (4.77 + 8.27i)4-s + (88.3 + 51.0i)5-s + (−62.3 + 223. i)6-s + (−127. + 318. i)7-s − 466. i·8-s + (−638. + 350. i)9-s + (437. + 757. i)10-s + (1.43e3 − 828. i)11-s + (−184. + 180. i)12-s − 278.·13-s + (−2.31e3 + 1.81e3i)14-s + (−741. + 2.65e3i)15-s + (2.30e3 − 3.99e3i)16-s + (4.95e3 − 2.85e3i)17-s + ⋯
L(s)  = 1  + (0.928 + 0.536i)2-s + (0.248 + 0.968i)3-s + (0.0746 + 0.129i)4-s + (0.706 + 0.408i)5-s + (−0.288 + 1.03i)6-s + (−0.372 + 0.928i)7-s − 0.911i·8-s + (−0.876 + 0.481i)9-s + (0.437 + 0.757i)10-s + (1.07 − 0.622i)11-s + (−0.106 + 0.104i)12-s − 0.126·13-s + (−0.843 + 0.662i)14-s + (−0.219 + 0.786i)15-s + (0.563 − 0.976i)16-s + (1.00 − 0.581i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0670 - 0.997i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.0670 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $0.0670 - 0.997i$
Analytic conductor: \(4.83113\)
Root analytic conductor: \(2.19798\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :3),\ 0.0670 - 0.997i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.89058 + 1.76775i\)
\(L(\frac12)\) \(\approx\) \(1.89058 + 1.76775i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-6.70 - 26.1i)T \)
7 \( 1 + (127. - 318. i)T \)
good2 \( 1 + (-7.42 - 4.28i)T + (32 + 55.4i)T^{2} \)
5 \( 1 + (-88.3 - 51.0i)T + (7.81e3 + 1.35e4i)T^{2} \)
11 \( 1 + (-1.43e3 + 828. i)T + (8.85e5 - 1.53e6i)T^{2} \)
13 \( 1 + 278.T + 4.82e6T^{2} \)
17 \( 1 + (-4.95e3 + 2.85e3i)T + (1.20e7 - 2.09e7i)T^{2} \)
19 \( 1 + (-2.93e3 + 5.08e3i)T + (-2.35e7 - 4.07e7i)T^{2} \)
23 \( 1 + (1.13e4 + 6.56e3i)T + (7.40e7 + 1.28e8i)T^{2} \)
29 \( 1 - 3.88e4iT - 5.94e8T^{2} \)
31 \( 1 + (-6.28e3 - 1.08e4i)T + (-4.43e8 + 7.68e8i)T^{2} \)
37 \( 1 + (3.81e4 - 6.60e4i)T + (-1.28e9 - 2.22e9i)T^{2} \)
41 \( 1 + 5.74e4iT - 4.75e9T^{2} \)
43 \( 1 + 2.51e3T + 6.32e9T^{2} \)
47 \( 1 + (4.16e4 + 2.40e4i)T + (5.38e9 + 9.33e9i)T^{2} \)
53 \( 1 + (2.17e4 - 1.25e4i)T + (1.10e10 - 1.91e10i)T^{2} \)
59 \( 1 + (3.45e4 - 1.99e4i)T + (2.10e10 - 3.65e10i)T^{2} \)
61 \( 1 + (7.33e4 - 1.27e5i)T + (-2.57e10 - 4.46e10i)T^{2} \)
67 \( 1 + (7.16e4 + 1.24e5i)T + (-4.52e10 + 7.83e10i)T^{2} \)
71 \( 1 + 6.19e5iT - 1.28e11T^{2} \)
73 \( 1 + (-2.54e5 - 4.40e5i)T + (-7.56e10 + 1.31e11i)T^{2} \)
79 \( 1 + (-1.61e5 + 2.79e5i)T + (-1.21e11 - 2.10e11i)T^{2} \)
83 \( 1 - 7.01e5iT - 3.26e11T^{2} \)
89 \( 1 + (6.30e5 + 3.63e5i)T + (2.48e11 + 4.30e11i)T^{2} \)
97 \( 1 - 1.47e6T + 8.32e11T^{2} \)
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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

−16.60039651201553933894693838993, −15.63541972137429925308247003093, −14.44945307654020475428483058707, −13.87226720600809752684106853691, −12.02872626536440272050423759544, −10.12239357111828637303970513415, −9.042136188396329614741866537660, −6.36356604414373061927019237143, −5.18331983146442969073001702603, −3.26409623777710071890110916184, 1.65474378334883015736286943806, 3.77829692897901767420541905464, 5.94950123685535469997615718577, 7.78883023189074434372952403490, 9.698025903231106194943811559312, 11.77696986268909393104888812432, 12.73039259704204675042359232911, 13.72763620858457907859424802262, 14.45776012426204124098711585620, 16.95355753490244618007371220055

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