Properties

Label 2-21-21.17-c1-0-0
Degree $2$
Conductor $21$
Sign $0.997 + 0.0633i$
Analytic cond. $0.167685$
Root an. cond. $0.409494$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.5 − 0.866i)3-s + (−1 + 1.73i)4-s + (0.5 − 2.59i)7-s + (1.5 + 2.59i)9-s + (3 − 1.73i)12-s + 1.73i·13-s + (−1.99 − 3.46i)16-s + (−4.5 + 2.59i)19-s + (−3 + 3.46i)21-s + (2.5 − 4.33i)25-s − 5.19i·27-s + (4 + 3.46i)28-s + (7.5 + 4.33i)31-s − 6·36-s + (−0.5 − 0.866i)37-s + ⋯
L(s)  = 1  + (−0.866 − 0.499i)3-s + (−0.5 + 0.866i)4-s + (0.188 − 0.981i)7-s + (0.5 + 0.866i)9-s + (0.866 − 0.499i)12-s + 0.480i·13-s + (−0.499 − 0.866i)16-s + (−1.03 + 0.596i)19-s + (−0.654 + 0.755i)21-s + (0.5 − 0.866i)25-s − 0.999i·27-s + (0.755 + 0.654i)28-s + (1.34 + 0.777i)31-s − 36-s + (−0.0821 − 0.142i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0633i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0633i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $0.997 + 0.0633i$
Analytic conductor: \(0.167685\)
Root analytic conductor: \(0.409494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :1/2),\ 0.997 + 0.0633i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.488931 - 0.0154982i\)
\(L(\frac12)\) \(\approx\) \(0.488931 - 0.0154982i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.5 + 0.866i)T \)
7 \( 1 + (-0.5 + 2.59i)T \)
good2 \( 1 + (1 - 1.73i)T^{2} \)
5 \( 1 + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 1.73iT - 13T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.5 - 2.59i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + (-7.5 - 4.33i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 5T + 43T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6 + 3.46i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.5 - 9.52i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (13.5 + 7.79i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.5 - 11.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 13.8iT - 97T^{2} \)
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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

−17.98512563028575517110355652782, −17.09957814568980073375252285183, −16.30300915821642460945913459265, −14.07800191195924878406018486295, −12.97159492547354371966127994182, −11.83685127878360589018873821263, −10.37061436856425034139712770950, −8.219434135148846900962796918633, −6.79345536931461537031444424174, −4.46412573016302803392609904678, 4.87678738065384911829484710659, 6.16523471638974682138318287764, 8.914387133758119867741938583574, 10.23518579966480607595970757355, 11.49613855591697476522057219719, 13.01262643877904661435148088220, 14.86571197707096909151568063946, 15.55152607378139954871197988100, 17.19890612710335839449793895734, 18.23347623226359451091631635260

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