Properties

Label 2-21-21.20-c21-0-12
Degree $2$
Conductor $21$
Sign $0.659 - 0.751i$
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  − 1.02e5i·3-s + 2.09e6·4-s + (−5.61e8 − 4.92e8i)7-s − 1.04e10·9-s − 2.14e11i·12-s + 9.22e11i·13-s + 4.39e12·16-s + 3.99e13i·19-s + (−5.03e13 + 5.74e13i)21-s − 4.76e14·25-s + 1.06e15i·27-s + (−1.17e15 − 1.03e15i)28-s − 1.25e15i·31-s − 2.19e16·36-s + 5.77e16·37-s + ⋯
L(s)  = 1  − 0.999i·3-s + 4-s + (−0.751 − 0.659i)7-s − 0.999·9-s − 0.999i·12-s + 1.85i·13-s + 16-s + 1.49i·19-s + (−0.659 + 0.751i)21-s − 0.999·25-s + 0.999i·27-s + (−0.751 − 0.659i)28-s − 0.275i·31-s − 0.999·36-s + 1.97·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.659 - 0.751i)\, \overline{\Lambda}(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & (0.659 - 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $0.659 - 0.751i$
Analytic conductor: \(58.6902\)
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: \((2,\ 21,\ (\ :21/2),\ 0.659 - 0.751i)\)

Particular Values

\(L(11)\) \(\approx\) \(1.587701918\)
\(L(\frac12)\) \(\approx\) \(1.587701918\)
\(L(\frac{23}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 1.02e5iT \)
7 \( 1 + (5.61e8 + 4.92e8i)T \)
good2 \( 1 - 2.09e6T^{2} \)
5 \( 1 + 4.76e14T^{2} \)
11 \( 1 - 7.40e21T^{2} \)
13 \( 1 - 9.22e11iT - 2.47e23T^{2} \)
17 \( 1 + 6.90e25T^{2} \)
19 \( 1 - 3.99e13iT - 7.14e26T^{2} \)
23 \( 1 - 3.94e28T^{2} \)
29 \( 1 - 5.13e30T^{2} \)
31 \( 1 + 1.25e15iT - 2.08e31T^{2} \)
37 \( 1 - 5.77e16T + 8.55e32T^{2} \)
41 \( 1 + 7.38e33T^{2} \)
43 \( 1 + 2.65e17T + 2.00e34T^{2} \)
47 \( 1 + 1.30e35T^{2} \)
53 \( 1 - 1.62e36T^{2} \)
59 \( 1 + 1.54e37T^{2} \)
61 \( 1 - 2.49e18iT - 3.10e37T^{2} \)
67 \( 1 - 6.94e18T + 2.22e38T^{2} \)
71 \( 1 - 7.52e38T^{2} \)
73 \( 1 - 6.21e19iT - 1.34e39T^{2} \)
79 \( 1 - 1.68e20T + 7.08e39T^{2} \)
83 \( 1 + 1.99e40T^{2} \)
89 \( 1 + 8.65e40T^{2} \)
97 \( 1 - 9.10e20iT - 5.27e41T^{2} \)
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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

−13.49569354382904190930906419692, −12.19200834271026321305495542484, −11.31691848112805478908525738983, −9.753470761907467476116248018365, −7.928099016901005355392369570320, −6.83550984800372671030370508226, −6.07978921998074230045655644739, −3.74850258106285630385815220731, −2.24979295557101562801861921486, −1.25900613417304301123181429083, 0.38331335069110227525029462216, 2.54615738994233115887787312201, 3.32841598919870425379346546950, 5.24498372983927034602958941826, 6.30491965200978824264409596211, 8.028631361168046090140950915120, 9.548860987763123891895986677996, 10.60840162707304358748483100720, 11.72013349629435612846346563264, 13.09818957488653745666804662523

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