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Properties

Label 2-21-3.2-c2-0-2
Degree $2$
Conductor $21$
Sign $0.607 + 0.794i$
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  − 1.30i·2-s + (−1.82 − 2.38i)3-s + 2.29·4-s + 7.37i·5-s + (−3.11 + 2.38i)6-s − 2.64·7-s − 8.22i·8-s + (−2.35 + 8.68i)9-s + 9.64·10-s − 2.61i·11-s + (−4.17 − 5.45i)12-s − 6.35·13-s + 3.45i·14-s + (17.5 − 13.4i)15-s − 1.58·16-s − 12.1i·17-s + ⋯
L(s)  = 1  − 0.653i·2-s + (−0.607 − 0.794i)3-s + 0.572·4-s + 1.47i·5-s + (−0.519 + 0.397i)6-s − 0.377·7-s − 1.02i·8-s + (−0.261 + 0.965i)9-s + 0.964·10-s − 0.237i·11-s + (−0.348 − 0.454i)12-s − 0.488·13-s + 0.247i·14-s + (1.17 − 0.896i)15-s − 0.0989·16-s − 0.714i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.607 + 0.794i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.607 + 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $0.607 + 0.794i$
Analytic conductor: \(0.572208\)
Root analytic conductor: \(0.756444\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :1),\ 0.607 + 0.794i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.742454 - 0.366798i\)
\(L(\frac12)\) \(\approx\) \(0.742454 - 0.366798i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.82 + 2.38i)T \)
7 \( 1 + 2.64T \)
good2 \( 1 + 1.30iT - 4T^{2} \)
5 \( 1 - 7.37iT - 25T^{2} \)
11 \( 1 + 2.61iT - 121T^{2} \)
13 \( 1 + 6.35T + 169T^{2} \)
17 \( 1 + 12.1iT - 289T^{2} \)
19 \( 1 + 10.2T + 361T^{2} \)
23 \( 1 - 4.30iT - 529T^{2} \)
29 \( 1 - 17.3iT - 841T^{2} \)
31 \( 1 - 39.2T + 961T^{2} \)
37 \( 1 - 41.0T + 1.36e3T^{2} \)
41 \( 1 + 30.2iT - 1.68e3T^{2} \)
43 \( 1 + 55.8T + 1.84e3T^{2} \)
47 \( 1 + 39.9iT - 2.20e3T^{2} \)
53 \( 1 - 105. iT - 2.80e3T^{2} \)
59 \( 1 - 41.3iT - 3.48e3T^{2} \)
61 \( 1 + 20.4T + 3.72e3T^{2} \)
67 \( 1 + 27.1T + 4.48e3T^{2} \)
71 \( 1 + 67.8iT - 5.04e3T^{2} \)
73 \( 1 - 60.7T + 5.32e3T^{2} \)
79 \( 1 + 63.2T + 6.24e3T^{2} \)
83 \( 1 - 89.9iT - 6.88e3T^{2} \)
89 \( 1 - 63.1iT - 7.92e3T^{2} \)
97 \( 1 - 19.1T + 9.40e3T^{2} \)
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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

−18.21700636530664691977716595000, −16.69668689372146167554062418988, −15.23636172881219556133467040338, −13.70199429714925516254534406272, −12.21212455621933931337508124739, −11.18926705230815417960428913541, −10.22044971433123207637602782910, −7.31850106587427905497581587199, −6.35514300323772687842844043143, −2.72394693730293101925304774111, 4.76189951599963393312237247570, 6.23009728629170269715395110239, 8.365753564601864572636346836896, 9.912622864366688966299283425911, 11.61055449222500752374842862982, 12.77676763769673097375325456987, 14.87537043484530243946296448027, 15.90918020796956462301426052915, 16.75110468915674496519824156456, 17.39330515913206427519390381782

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