Properties

Label 2-21-7.3-c6-0-3
Degree $2$
Conductor $21$
Sign $-0.659 - 0.751i$
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  + (6.58 + 11.4i)2-s + (13.5 + 7.79i)3-s + (−54.8 + 94.9i)4-s + (−68.9 + 39.7i)5-s + 205. i·6-s + (284. − 191. i)7-s − 601.·8-s + (121.5 + 210. i)9-s + (−908. − 524. i)10-s + (−411. + 712. i)11-s + (−1.48e3 + 854. i)12-s − 2.42e3i·13-s + (4.06e3 + 1.97e3i)14-s − 1.24e3·15-s + (−456. − 791. i)16-s + (6.75e3 + 3.89e3i)17-s + ⋯
L(s)  = 1  + (0.823 + 1.42i)2-s + (0.5 + 0.288i)3-s + (−0.856 + 1.48i)4-s + (−0.551 + 0.318i)5-s + 0.951i·6-s + (0.828 − 0.559i)7-s − 1.17·8-s + (0.166 + 0.288i)9-s + (−0.908 − 0.524i)10-s + (−0.308 + 0.534i)11-s + (−0.856 + 0.494i)12-s − 1.10i·13-s + (1.48 + 0.721i)14-s − 0.367·15-s + (−0.111 − 0.193i)16-s + (1.37 + 0.793i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.659 - 0.751i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.659 - 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $-0.659 - 0.751i$
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: \((2,\ 21,\ (\ :3),\ -0.659 - 0.751i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.01761 + 2.24836i\)
\(L(\frac12)\) \(\approx\) \(1.01761 + 2.24836i\)
\(L(4)\) 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 \( 1 + (-284. + 191. i)T \)
good2 \( 1 + (-6.58 - 11.4i)T + (-32 + 55.4i)T^{2} \)
5 \( 1 + (68.9 - 39.7i)T + (7.81e3 - 1.35e4i)T^{2} \)
11 \( 1 + (411. - 712. i)T + (-8.85e5 - 1.53e6i)T^{2} \)
13 \( 1 + 2.42e3iT - 4.82e6T^{2} \)
17 \( 1 + (-6.75e3 - 3.89e3i)T + (1.20e7 + 2.09e7i)T^{2} \)
19 \( 1 + (-5.78e3 + 3.34e3i)T + (2.35e7 - 4.07e7i)T^{2} \)
23 \( 1 + (9.41e3 + 1.63e4i)T + (-7.40e7 + 1.28e8i)T^{2} \)
29 \( 1 - 1.38e4T + 5.94e8T^{2} \)
31 \( 1 + (2.41e4 + 1.39e4i)T + (4.43e8 + 7.68e8i)T^{2} \)
37 \( 1 + (3.98e4 + 6.90e4i)T + (-1.28e9 + 2.22e9i)T^{2} \)
41 \( 1 + 5.91e4iT - 4.75e9T^{2} \)
43 \( 1 + 9.18e4T + 6.32e9T^{2} \)
47 \( 1 + (4.34e3 - 2.50e3i)T + (5.38e9 - 9.33e9i)T^{2} \)
53 \( 1 + (9.31e4 - 1.61e5i)T + (-1.10e10 - 1.91e10i)T^{2} \)
59 \( 1 + (-1.95e5 - 1.12e5i)T + (2.10e10 + 3.65e10i)T^{2} \)
61 \( 1 + (1.25e5 - 7.21e4i)T + (2.57e10 - 4.46e10i)T^{2} \)
67 \( 1 + (-1.17e5 + 2.03e5i)T + (-4.52e10 - 7.83e10i)T^{2} \)
71 \( 1 - 9.62e4T + 1.28e11T^{2} \)
73 \( 1 + (-2.38e5 - 1.37e5i)T + (7.56e10 + 1.31e11i)T^{2} \)
79 \( 1 + (-3.40e5 - 5.90e5i)T + (-1.21e11 + 2.10e11i)T^{2} \)
83 \( 1 - 1.28e5iT - 3.26e11T^{2} \)
89 \( 1 + (3.22e5 - 1.86e5i)T + (2.48e11 - 4.30e11i)T^{2} \)
97 \( 1 + 6.20e5iT - 8.32e11T^{2} \)
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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

−16.89191782623276427306316603236, −15.64175250067608447735542028359, −14.82039606503355772688714888834, −13.98857684243388162070090690347, −12.53606873204608294989904815316, −10.48065601427312615298428259008, −8.127661995683913136231997827509, −7.39370344282558503386230958903, −5.31448850904758954752247635665, −3.79044233514880082465761248122, 1.55022349644524571884962061999, 3.39924450791436126773001582690, 5.12585294616413993570879270517, 8.019499986775679808237289848000, 9.736210376158986569242947480114, 11.62851439682204977024596011979, 12.03103382927762455701678215747, 13.68225643894510232533374683503, 14.43489410169520588774001579603, 16.12564156860900616670680061863

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