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Properties

Label 2-21-21.20-c21-0-27
Degree $2$
Conductor $21$
Sign $0.723 + 0.690i$
Analytic cond. $58.6902$
Root an. cond. $7.66095$
Motivic weight $21$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.22e3i·2-s + (9.40e4 + 4.02e4i)3-s + 5.99e5·4-s − 3.45e7·5-s + (4.92e7 − 1.15e8i)6-s + (−6.99e8 − 2.61e8i)7-s − 3.30e9i·8-s + (7.22e9 + 7.56e9i)9-s + 4.22e10i·10-s + 1.38e11i·11-s + (5.63e10 + 2.41e10i)12-s − 1.98e11i·13-s + (−3.20e11 + 8.56e11i)14-s + (−3.24e12 − 1.39e12i)15-s − 2.78e12·16-s + 1.12e13·17-s + ⋯
L(s)  = 1  − 0.845i·2-s + (0.919 + 0.393i)3-s + 0.285·4-s − 1.58·5-s + (0.332 − 0.776i)6-s + (−0.936 − 0.350i)7-s − 1.08i·8-s + (0.690 + 0.723i)9-s + 1.33i·10-s + 1.60i·11-s + (0.262 + 0.112i)12-s − 0.399i·13-s + (−0.296 + 0.791i)14-s + (−1.45 − 0.622i)15-s − 0.632·16-s + 1.34·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.723 + 0.690i)\, \overline{\Lambda}(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & (0.723 + 0.690i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $0.723 + 0.690i$
Analytic conductor: \(58.6902\)
Root analytic conductor: \(7.66095\)
Motivic weight: \(21\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :21/2),\ 0.723 + 0.690i)\)

Particular Values

\(L(11)\) \(\approx\) \(2.316987672\)
\(L(\frac12)\) \(\approx\) \(2.316987672\)
\(L(\frac{23}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-9.40e4 - 4.02e4i)T \)
7 \( 1 + (6.99e8 + 2.61e8i)T \)
good2 \( 1 + 1.22e3iT - 2.09e6T^{2} \)
5 \( 1 + 3.45e7T + 4.76e14T^{2} \)
11 \( 1 - 1.38e11iT - 7.40e21T^{2} \)
13 \( 1 + 1.98e11iT - 2.47e23T^{2} \)
17 \( 1 - 1.12e13T + 6.90e25T^{2} \)
19 \( 1 - 1.31e13iT - 7.14e26T^{2} \)
23 \( 1 + 2.41e14iT - 3.94e28T^{2} \)
29 \( 1 + 2.94e15iT - 5.13e30T^{2} \)
31 \( 1 - 5.22e15iT - 2.08e31T^{2} \)
37 \( 1 - 1.95e16T + 8.55e32T^{2} \)
41 \( 1 - 8.86e16T + 7.38e33T^{2} \)
43 \( 1 + 2.83e16T + 2.00e34T^{2} \)
47 \( 1 - 5.37e17T + 1.30e35T^{2} \)
53 \( 1 + 1.93e18iT - 1.62e36T^{2} \)
59 \( 1 - 1.38e18T + 1.54e37T^{2} \)
61 \( 1 - 2.80e18iT - 3.10e37T^{2} \)
67 \( 1 - 1.55e19T + 2.22e38T^{2} \)
71 \( 1 + 1.31e19iT - 7.52e38T^{2} \)
73 \( 1 - 3.15e19iT - 1.34e39T^{2} \)
79 \( 1 + 6.56e18T + 7.08e39T^{2} \)
83 \( 1 - 1.99e20T + 1.99e40T^{2} \)
89 \( 1 + 7.23e19T + 8.65e40T^{2} \)
97 \( 1 + 6.71e19iT - 5.27e41T^{2} \)
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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

−12.74690251482246982184510998824, −12.13706023241519547071323931866, −10.50540428372162052390982454226, −9.722714783185882691592708924400, −7.914827795656051162597336810165, −7.04468486052478643810743874925, −4.27978982112521432458157000393, −3.53120655335588831884657760594, −2.46769765214698930113657348326, −0.78114543526072205669105616672, 0.789371458192486028441515314867, 2.87052706926643366901285547914, 3.68038898946948753867514818490, 5.88118511549259607431432276563, 7.21578326788314911504072103900, 7.989852446558125008966064583103, 9.063547551634622465269685722398, 11.23464271182187228561546116454, 12.31261443596583677272463436796, 13.84575663299844360767294801159

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