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Properties

Label 2-21-21.11-c2-0-2
Degree $2$
Conductor $21$
Sign $0.928 + 0.371i$
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  + (1.93 − 1.11i)2-s + (−2.93 + 0.614i)3-s + (0.5 − 0.866i)4-s + (−1.93 + 1.11i)5-s + (−5.00 + 4.47i)6-s + (3.5 − 6.06i)7-s + 6.70i·8-s + (8.24 − 3.60i)9-s + (−2.5 + 4.33i)10-s + (−9.68 − 5.59i)11-s + (−0.936 + 2.85i)12-s − 2·13-s − 15.6i·14-s + (5.00 − 4.47i)15-s + (9.5 + 16.4i)16-s + (23.2 + 13.4i)17-s + ⋯
L(s)  = 1  + (0.968 − 0.559i)2-s + (−0.978 + 0.204i)3-s + (0.125 − 0.216i)4-s + (−0.387 + 0.223i)5-s + (−0.833 + 0.745i)6-s + (0.5 − 0.866i)7-s + 0.838i·8-s + (0.916 − 0.400i)9-s + (−0.250 + 0.433i)10-s + (−0.880 − 0.508i)11-s + (−0.0780 + 0.237i)12-s − 0.153·13-s − 1.11i·14-s + (0.333 − 0.298i)15-s + (0.593 + 1.02i)16-s + (1.36 + 0.789i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.928 + 0.371i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.928 + 0.371i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $0.928 + 0.371i$
Analytic conductor: \(0.572208\)
Root analytic conductor: \(0.756444\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :1),\ 0.928 + 0.371i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.992942 - 0.191397i\)
\(L(\frac12)\) \(\approx\) \(0.992942 - 0.191397i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.93 - 0.614i)T \)
7 \( 1 + (-3.5 + 6.06i)T \)
good2 \( 1 + (-1.93 + 1.11i)T + (2 - 3.46i)T^{2} \)
5 \( 1 + (1.93 - 1.11i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (9.68 + 5.59i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + 2T + 169T^{2} \)
17 \( 1 + (-23.2 - 13.4i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (8 + 13.8i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (11.6 - 6.70i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + 15.6iT - 841T^{2} \)
31 \( 1 + (-1.5 + 2.59i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (6 + 10.3i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 31.3iT - 1.68e3T^{2} \)
43 \( 1 - 44T + 1.84e3T^{2} \)
47 \( 1 + (-11.6 + 6.70i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-17.4 - 10.0i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (17.4 + 10.0i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-13 - 22.5i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (26 - 45.0i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 93.9iT - 5.04e3T^{2} \)
73 \( 1 + (9 - 15.5i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-39.5 - 68.4i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 140. iT - 6.88e3T^{2} \)
89 \( 1 + (42.6 - 24.5i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 93T + 9.40e3T^{2} \)
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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

−17.70849544448348399096742239171, −16.75070961773870300137640036501, −15.22146390234541627482057730647, −13.78091377295046842559865238585, −12.56380924719453118065801914145, −11.36073541210854960998510300557, −10.47036081354431830917804358796, −7.73141960301047997538788715575, −5.48903860934316606884924599157, −3.94542690833329969833691836104, 4.77249049080509833323078401705, 5.86317044138694108514974959027, 7.66361245564424492733683551836, 10.11171694228189508306715071413, 11.96745805139525695531750575152, 12.69062168776067391165718117538, 14.32389765542925139278161845480, 15.56618379942246307775577258454, 16.41054882307387875101483910467, 18.08321292704430157315563847028

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