Properties

Label 2-21-7.3-c6-0-5
Degree $2$
Conductor $21$
Sign $0.844 + 0.535i$
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  + (2.25 + 3.90i)2-s + (−13.5 − 7.79i)3-s + (21.8 − 37.8i)4-s + (53.9 − 31.1i)5-s − 70.2i·6-s + (218. − 264. i)7-s + 485.·8-s + (121.5 + 210. i)9-s + (243. + 140. i)10-s + (−9.71 + 16.8i)11-s + (−589. + 340. i)12-s − 1.64e3i·13-s + (1.52e3 + 258. i)14-s − 970.·15-s + (−304. − 527. i)16-s + (−186. − 107. i)17-s + ⋯
L(s)  = 1  + (0.281 + 0.487i)2-s + (−0.5 − 0.288i)3-s + (0.341 − 0.591i)4-s + (0.431 − 0.249i)5-s − 0.325i·6-s + (0.637 − 0.770i)7-s + 0.947·8-s + (0.166 + 0.288i)9-s + (0.243 + 0.140i)10-s + (−0.00730 + 0.0126i)11-s + (−0.341 + 0.197i)12-s − 0.747i·13-s + (0.555 + 0.0943i)14-s − 0.287·15-s + (−0.0743 − 0.128i)16-s + (−0.0378 − 0.0218i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.844 + 0.535i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.844 + 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $0.844 + 0.535i$
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: \((2,\ 21,\ (\ :3),\ 0.844 + 0.535i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.74293 - 0.506042i\)
\(L(\frac12)\) \(\approx\) \(1.74293 - 0.506042i\)
\(L(4)\) 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 \( 1 + (-218. + 264. i)T \)
good2 \( 1 + (-2.25 - 3.90i)T + (-32 + 55.4i)T^{2} \)
5 \( 1 + (-53.9 + 31.1i)T + (7.81e3 - 1.35e4i)T^{2} \)
11 \( 1 + (9.71 - 16.8i)T + (-8.85e5 - 1.53e6i)T^{2} \)
13 \( 1 + 1.64e3iT - 4.82e6T^{2} \)
17 \( 1 + (186. + 107. i)T + (1.20e7 + 2.09e7i)T^{2} \)
19 \( 1 + (8.23e3 - 4.75e3i)T + (2.35e7 - 4.07e7i)T^{2} \)
23 \( 1 + (-7.22e3 - 1.25e4i)T + (-7.40e7 + 1.28e8i)T^{2} \)
29 \( 1 - 4.30e4T + 5.94e8T^{2} \)
31 \( 1 + (7.79e3 + 4.50e3i)T + (4.43e8 + 7.68e8i)T^{2} \)
37 \( 1 + (-1.65e4 - 2.86e4i)T + (-1.28e9 + 2.22e9i)T^{2} \)
41 \( 1 - 7.37e4iT - 4.75e9T^{2} \)
43 \( 1 - 4.76e3T + 6.32e9T^{2} \)
47 \( 1 + (6.37e4 - 3.68e4i)T + (5.38e9 - 9.33e9i)T^{2} \)
53 \( 1 + (1.19e5 - 2.06e5i)T + (-1.10e10 - 1.91e10i)T^{2} \)
59 \( 1 + (2.82e5 + 1.63e5i)T + (2.10e10 + 3.65e10i)T^{2} \)
61 \( 1 + (-3.50e5 + 2.02e5i)T + (2.57e10 - 4.46e10i)T^{2} \)
67 \( 1 + (1.09e5 - 1.90e5i)T + (-4.52e10 - 7.83e10i)T^{2} \)
71 \( 1 - 3.50e5T + 1.28e11T^{2} \)
73 \( 1 + (1.74e5 + 1.00e5i)T + (7.56e10 + 1.31e11i)T^{2} \)
79 \( 1 + (1.97e5 + 3.42e5i)T + (-1.21e11 + 2.10e11i)T^{2} \)
83 \( 1 + 1.32e4iT - 3.26e11T^{2} \)
89 \( 1 + (1.99e5 - 1.15e5i)T + (2.48e11 - 4.30e11i)T^{2} \)
97 \( 1 - 6.62e5iT - 8.32e11T^{2} \)
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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

−16.80706422117315366277221329283, −15.41043937253150549749790615314, −14.17646402019499483269608869929, −13.03368594729163863519933628286, −11.21769898977934792588740280263, −10.13400110489660328569142334283, −7.79335895041411157492651736453, −6.28310245887275629856488599191, −4.89275420391142568589887323571, −1.34389304503324237964612702304, 2.32132558103271108683911239374, 4.57774771345816920987023905309, 6.58246856479992647886511103147, 8.614783788116312218292973789187, 10.55372207454389829572838857749, 11.62699207873562796061335268639, 12.70223651668418656558681167741, 14.30317049927444420975997696890, 15.76738470741456296165798996725, 17.01375784434937517610302495952

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