Properties

Label 2-21-21.11-c6-0-8
Degree $2$
Conductor $21$
Sign $0.772 + 0.635i$
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  + (−2.88 + 1.66i)2-s + (−26.7 − 4.00i)3-s + (−26.4 + 45.8i)4-s + (166. − 96.0i)5-s + (83.7 − 32.9i)6-s + (157. − 304. i)7-s − 389. i·8-s + (696. + 214. i)9-s + (−319. + 554. i)10-s + (−750. − 433. i)11-s + (889. − 1.11e3i)12-s + 3.41e3·13-s + (54.6 + 1.14e3i)14-s + (−4.82e3 + 1.89e3i)15-s + (−1.04e3 − 1.80e3i)16-s + (−2.06e3 − 1.19e3i)17-s + ⋯
L(s)  = 1  + (−0.360 + 0.208i)2-s + (−0.988 − 0.148i)3-s + (−0.413 + 0.715i)4-s + (1.33 − 0.768i)5-s + (0.387 − 0.152i)6-s + (0.457 − 0.888i)7-s − 0.760i·8-s + (0.955 + 0.293i)9-s + (−0.319 + 0.554i)10-s + (−0.564 − 0.325i)11-s + (0.514 − 0.646i)12-s + 1.55·13-s + (0.0199 + 0.415i)14-s + (−1.43 + 0.562i)15-s + (−0.254 − 0.441i)16-s + (−0.420 − 0.242i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.772 + 0.635i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.772 + 0.635i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $0.772 + 0.635i$
Analytic conductor: \(4.83113\)
Root analytic conductor: \(2.19798\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :3),\ 0.772 + 0.635i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.981837 - 0.351816i\)
\(L(\frac12)\) \(\approx\) \(0.981837 - 0.351816i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (26.7 + 4.00i)T \)
7 \( 1 + (-157. + 304. i)T \)
good2 \( 1 + (2.88 - 1.66i)T + (32 - 55.4i)T^{2} \)
5 \( 1 + (-166. + 96.0i)T + (7.81e3 - 1.35e4i)T^{2} \)
11 \( 1 + (750. + 433. i)T + (8.85e5 + 1.53e6i)T^{2} \)
13 \( 1 - 3.41e3T + 4.82e6T^{2} \)
17 \( 1 + (2.06e3 + 1.19e3i)T + (1.20e7 + 2.09e7i)T^{2} \)
19 \( 1 + (1.29e3 + 2.24e3i)T + (-2.35e7 + 4.07e7i)T^{2} \)
23 \( 1 + (-6.49e3 + 3.74e3i)T + (7.40e7 - 1.28e8i)T^{2} \)
29 \( 1 + 2.45e4iT - 5.94e8T^{2} \)
31 \( 1 + (-1.02e4 + 1.76e4i)T + (-4.43e8 - 7.68e8i)T^{2} \)
37 \( 1 + (-3.39e4 - 5.88e4i)T + (-1.28e9 + 2.22e9i)T^{2} \)
41 \( 1 - 8.76e4iT - 4.75e9T^{2} \)
43 \( 1 - 1.12e4T + 6.32e9T^{2} \)
47 \( 1 + (1.05e5 - 6.11e4i)T + (5.38e9 - 9.33e9i)T^{2} \)
53 \( 1 + (1.45e5 + 8.37e4i)T + (1.10e10 + 1.91e10i)T^{2} \)
59 \( 1 + (-1.39e5 - 8.02e4i)T + (2.10e10 + 3.65e10i)T^{2} \)
61 \( 1 + (-1.99e4 - 3.45e4i)T + (-2.57e10 + 4.46e10i)T^{2} \)
67 \( 1 + (-3.69e4 + 6.39e4i)T + (-4.52e10 - 7.83e10i)T^{2} \)
71 \( 1 + 9.29e3iT - 1.28e11T^{2} \)
73 \( 1 + (7.80e4 - 1.35e5i)T + (-7.56e10 - 1.31e11i)T^{2} \)
79 \( 1 + (1.16e5 + 2.02e5i)T + (-1.21e11 + 2.10e11i)T^{2} \)
83 \( 1 - 8.72e5iT - 3.26e11T^{2} \)
89 \( 1 + (-1.82e5 + 1.05e5i)T + (2.48e11 - 4.30e11i)T^{2} \)
97 \( 1 - 1.18e6T + 8.32e11T^{2} \)
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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

−16.95941553889843956845735512569, −16.09648874874367807690202612495, −13.34343406765518834582722307677, −13.24303247901064334414573700291, −11.22678560369360850283511754799, −9.784390181685032084731520772826, −8.214648399784020659666306923682, −6.34770378260261294891410016590, −4.64795125404776215419752749480, −0.955308097150233565553781510414, 1.71559517028066329068413388382, 5.32578453503094994540844992058, 6.25694393758566796844834322860, 9.029769290518067725232307891872, 10.37015921885757294563317998600, 11.10869547947453536117732298077, 13.06967932956724525539323604018, 14.43379104249514379008628075020, 15.69897661937748624910631020702, 17.57092072168741759407546412450

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