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Properties

Label 2-21-21.20-c21-0-30
Degree $2$
Conductor $21$
Sign $-0.138 - 0.990i$
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  + 247. i·2-s + (−1.62e4 + 1.00e5i)3-s + 2.03e6·4-s + 3.48e7·5-s + (−2.49e7 − 4.00e6i)6-s + (7.14e8 − 2.19e8i)7-s + 1.02e9i·8-s + (−9.93e9 − 3.27e9i)9-s + 8.59e9i·10-s + 9.21e10i·11-s + (−3.30e10 + 2.05e11i)12-s + 1.67e11i·13-s + (5.41e10 + 1.76e11i)14-s + (−5.64e11 + 3.51e12i)15-s + 4.01e12·16-s − 7.92e12·17-s + ⋯
L(s)  = 1  + 0.170i·2-s + (−0.158 + 0.987i)3-s + 0.970·4-s + 1.59·5-s + (−0.168 − 0.0270i)6-s + (0.955 − 0.293i)7-s + 0.336i·8-s + (−0.949 − 0.312i)9-s + 0.271i·10-s + 1.07i·11-s + (−0.153 + 0.958i)12-s + 0.337i·13-s + (0.0500 + 0.163i)14-s + (−0.252 + 1.57i)15-s + 0.913·16-s − 0.953·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.138 - 0.990i)\, \overline{\Lambda}(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & (-0.138 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $-0.138 - 0.990i$
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: \((2,\ 21,\ (\ :21/2),\ -0.138 - 0.990i)\)

Particular Values

\(L(11)\) \(\approx\) \(3.835170821\)
\(L(\frac12)\) \(\approx\) \(3.835170821\)
\(L(\frac{23}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.62e4 - 1.00e5i)T \)
7 \( 1 + (-7.14e8 + 2.19e8i)T \)
good2 \( 1 - 247. iT - 2.09e6T^{2} \)
5 \( 1 - 3.48e7T + 4.76e14T^{2} \)
11 \( 1 - 9.21e10iT - 7.40e21T^{2} \)
13 \( 1 - 1.67e11iT - 2.47e23T^{2} \)
17 \( 1 + 7.92e12T + 6.90e25T^{2} \)
19 \( 1 - 3.32e13iT - 7.14e26T^{2} \)
23 \( 1 + 3.78e14iT - 3.94e28T^{2} \)
29 \( 1 - 2.17e15iT - 5.13e30T^{2} \)
31 \( 1 - 3.73e15iT - 2.08e31T^{2} \)
37 \( 1 + 1.35e16T + 8.55e32T^{2} \)
41 \( 1 - 6.54e16T + 7.38e33T^{2} \)
43 \( 1 - 4.14e15T + 2.00e34T^{2} \)
47 \( 1 - 4.66e17T + 1.30e35T^{2} \)
53 \( 1 + 1.25e18iT - 1.62e36T^{2} \)
59 \( 1 + 5.49e17T + 1.54e37T^{2} \)
61 \( 1 - 7.00e18iT - 3.10e37T^{2} \)
67 \( 1 + 1.30e19T + 2.22e38T^{2} \)
71 \( 1 + 3.38e19iT - 7.52e38T^{2} \)
73 \( 1 - 5.69e19iT - 1.34e39T^{2} \)
79 \( 1 + 5.19e19T + 7.08e39T^{2} \)
83 \( 1 + 1.77e20T + 1.99e40T^{2} \)
89 \( 1 - 4.40e20T + 8.65e40T^{2} \)
97 \( 1 + 5.59e20iT - 5.27e41T^{2} \)
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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

−14.22078197433375869430624151674, −12.30981684450548205882739744877, −10.77386206277472597946571117330, −10.16719474643301018548687180836, −8.709778215957514501127051093873, −6.83315830772204777639830604440, −5.65985627266955150634830702842, −4.49696603045563142920385575009, −2.46900358901030569212491593795, −1.58299732251818925173105210300, 0.941115617212594696554496756152, 1.93701768467226510224130068243, 2.68749020253957738075732413181, 5.50081844878382731782417143028, 6.20294970159826119629156622020, 7.56820173961638667315127134293, 9.071520638102200931282715472012, 10.85666658003504330190874013478, 11.60867225933170707276709940111, 13.20630504641479562024637536908

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