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Properties

Label 2-21-7.5-c2-0-1
Degree $2$
Conductor $21$
Sign $0.922 - 0.386i$
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  + (−0.5 + 0.866i)2-s + (1.5 − 0.866i)3-s + (1.5 + 2.59i)4-s + (−4.5 − 2.59i)5-s + 1.73i·6-s + (−3.5 − 6.06i)7-s − 7·8-s + (1.5 − 2.59i)9-s + (4.5 − 2.59i)10-s + (5.5 + 9.52i)11-s + (4.5 + 2.59i)12-s + 6.92i·13-s + 7·14-s − 9·15-s + (−2.5 + 4.33i)16-s + (21 − 12.1i)17-s + ⋯
L(s)  = 1  + (−0.250 + 0.433i)2-s + (0.5 − 0.288i)3-s + (0.375 + 0.649i)4-s + (−0.900 − 0.519i)5-s + 0.288i·6-s + (−0.5 − 0.866i)7-s − 0.875·8-s + (0.166 − 0.288i)9-s + (0.450 − 0.259i)10-s + (0.5 + 0.866i)11-s + (0.375 + 0.216i)12-s + 0.532i·13-s + 0.5·14-s − 0.599·15-s + (−0.156 + 0.270i)16-s + (1.23 − 0.713i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.922 - 0.386i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.922 - 0.386i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $0.922 - 0.386i$
Analytic conductor: \(0.572208\)
Root analytic conductor: \(0.756444\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :1),\ 0.922 - 0.386i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.824806 + 0.165843i\)
\(L(\frac12)\) \(\approx\) \(0.824806 + 0.165843i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.5 + 0.866i)T \)
7 \( 1 + (3.5 + 6.06i)T \)
good2 \( 1 + (0.5 - 0.866i)T + (-2 - 3.46i)T^{2} \)
5 \( 1 + (4.5 + 2.59i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (-5.5 - 9.52i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 - 6.92iT - 169T^{2} \)
17 \( 1 + (-21 + 12.1i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (3 + 1.73i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (14 - 24.2i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 - 25T + 841T^{2} \)
31 \( 1 + (28.5 - 16.4i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-29 + 50.2i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 3.46iT - 1.68e3T^{2} \)
43 \( 1 - 26T + 1.84e3T^{2} \)
47 \( 1 + (66 + 38.1i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (15.5 + 26.8i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (7.5 - 4.33i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-12 - 6.92i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-26 - 45.0i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 64T + 5.04e3T^{2} \)
73 \( 1 + (-6 + 3.46i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (8.5 - 14.7i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 53.6iT - 6.88e3T^{2} \)
89 \( 1 + (69 + 39.8i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 91.7iT - 9.40e3T^{2} \)
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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

−17.87713402532611485950052584663, −16.59476181623643885669766982873, −15.85860717688117591062652482968, −14.34633160381525503452410563419, −12.70873457819104802394232496202, −11.75592968208496928270867203428, −9.494130765849989756054432642969, −7.910717366185993546844553523571, −7.00377100269439949615422199973, −3.80768540697067178356210859712, 3.17648036595521212323167511635, 6.11761929321551634880283495148, 8.269149256054203646871253978662, 9.820437134345424814553307718775, 11.11304553140246309728494652812, 12.34463758149409219845136168174, 14.46196494531299827678109479040, 15.25478531882660163329757668504, 16.32490042475320846356153975606, 18.57487378858281785048697742474

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