Properties

Label 2-21-7.6-c2-0-1
Degree $2$
Conductor $21$
Sign $0.989 + 0.142i$
Analytic cond. $0.572208$
Root an. cond. $0.756444$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 1.73i·3-s − 3·4-s + 6.92i·5-s − 1.73i·6-s + (1 − 6.92i)7-s − 7·8-s − 2.99·9-s + 6.92i·10-s + 10·11-s + 5.19i·12-s − 6.92i·13-s + (1 − 6.92i)14-s + 11.9·15-s + 5·16-s + ⋯
L(s)  = 1  + 0.5·2-s − 0.577i·3-s − 0.750·4-s + 1.38i·5-s − 0.288i·6-s + (0.142 − 0.989i)7-s − 0.875·8-s − 0.333·9-s + 0.692i·10-s + 0.909·11-s + 0.433i·12-s − 0.532i·13-s + (0.0714 − 0.494i)14-s + 0.799·15-s + 0.312·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.142i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.989 + 0.142i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $0.989 + 0.142i$
Analytic conductor: \(0.572208\)
Root analytic conductor: \(0.756444\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :1),\ 0.989 + 0.142i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.952768 - 0.0684057i\)
\(L(\frac12)\) \(\approx\) \(0.952768 - 0.0684057i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 1.73iT \)
7 \( 1 + (-1 + 6.92i)T \)
good2 \( 1 - T + 4T^{2} \)
5 \( 1 - 6.92iT - 25T^{2} \)
11 \( 1 - 10T + 121T^{2} \)
13 \( 1 + 6.92iT - 169T^{2} \)
17 \( 1 - 289T^{2} \)
19 \( 1 - 20.7iT - 361T^{2} \)
23 \( 1 + 14T + 529T^{2} \)
29 \( 1 + 38T + 841T^{2} \)
31 \( 1 + 27.7iT - 961T^{2} \)
37 \( 1 - 26T + 1.36e3T^{2} \)
41 \( 1 - 69.2iT - 1.68e3T^{2} \)
43 \( 1 - 26T + 1.84e3T^{2} \)
47 \( 1 + 27.7iT - 2.20e3T^{2} \)
53 \( 1 - 10T + 2.80e3T^{2} \)
59 \( 1 + 76.2iT - 3.48e3T^{2} \)
61 \( 1 + 34.6iT - 3.72e3T^{2} \)
67 \( 1 - 74T + 4.48e3T^{2} \)
71 \( 1 + 62T + 5.04e3T^{2} \)
73 \( 1 - 41.5iT - 5.32e3T^{2} \)
79 \( 1 + 46T + 6.24e3T^{2} \)
83 \( 1 + 90.0iT - 6.88e3T^{2} \)
89 \( 1 - 41.5iT - 7.92e3T^{2} \)
97 \( 1 + 55.4iT - 9.40e3T^{2} \)
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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

−18.10517069117350486220117120460, −16.99709269874029433437998324888, −14.75742959313816807996059400352, −14.19899250884021282252848878322, −13.01727907733575660680791099605, −11.40828011099156541309646669122, −9.904220807212029954536797316677, −7.76342273456707659248606363139, −6.22659888357279980443063344171, −3.71728929475028991669282655984, 4.31814990297491096689755315369, 5.56031607780747393425907988408, 8.795882945302853429926271393819, 9.278044820705301326040810468881, 11.76131059760542062142366906094, 12.77709226594676102208899896857, 14.14940590373736655263465816724, 15.41901003723195167156855700492, 16.68516973657824329419909365056, 17.81695983604413400532376722810

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