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Properties

Label 2-21-7.3-c2-0-2
Degree $2$
Conductor $21$
Sign $-0.250 + 0.968i$
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  + (−1.5 − 2.59i)2-s + (−1.5 − 0.866i)3-s + (−2.5 + 4.33i)4-s + (4.5 − 2.59i)5-s + 5.19i·6-s + (6.5 + 2.59i)7-s + 3.00·8-s + (1.5 + 2.59i)9-s + (−13.5 − 7.79i)10-s + (−7.5 + 12.9i)11-s + (7.5 − 4.33i)12-s − 13.8i·13-s + (−3 − 20.7i)14-s − 9·15-s + (5.49 + 9.52i)16-s + (9 + 5.19i)17-s + ⋯
L(s)  = 1  + (−0.750 − 1.29i)2-s + (−0.5 − 0.288i)3-s + (−0.625 + 1.08i)4-s + (0.900 − 0.519i)5-s + 0.866i·6-s + (0.928 + 0.371i)7-s + 0.375·8-s + (0.166 + 0.288i)9-s + (−1.35 − 0.779i)10-s + (−0.681 + 1.18i)11-s + (0.625 − 0.360i)12-s − 1.06i·13-s + (−0.214 − 1.48i)14-s − 0.599·15-s + (0.343 + 0.595i)16-s + (0.529 + 0.305i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.250 + 0.968i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.250 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $-0.250 + 0.968i$
Analytic conductor: \(0.572208\)
Root analytic conductor: \(0.756444\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :1),\ -0.250 + 0.968i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.383309 - 0.495269i\)
\(L(\frac12)\) \(\approx\) \(0.383309 - 0.495269i\)
\(L(2)\) 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 \( 1 + (-6.5 - 2.59i)T \)
good2 \( 1 + (1.5 + 2.59i)T + (-2 + 3.46i)T^{2} \)
5 \( 1 + (-4.5 + 2.59i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (7.5 - 12.9i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + 13.8iT - 169T^{2} \)
17 \( 1 + (-9 - 5.19i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (9 - 5.19i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (-264.5 + 458. i)T^{2} \)
29 \( 1 + 9T + 841T^{2} \)
31 \( 1 + (10.5 + 6.06i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (5 + 8.66i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 10.3iT - 1.68e3T^{2} \)
43 \( 1 + 74T + 1.84e3T^{2} \)
47 \( 1 + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (16.5 - 28.5i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-13.5 - 7.79i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-78 + 45.0i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-38 + 65.8i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 84T + 5.04e3T^{2} \)
73 \( 1 + (54 + 31.1i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-21.5 - 37.2i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 119. iT - 6.88e3T^{2} \)
89 \( 1 + (63 - 36.3i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 185. iT - 9.40e3T^{2} \)
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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

−17.80953954122890582278245943390, −17.18327266549577149604830218989, −15.05999612091091947481807743981, −13.03946601412904954172944165990, −12.21360208699840445746088239509, −10.76218975895576606865220093901, −9.737276605148152447619533187655, −8.117815582577185859987229594189, −5.38247025637035665847715214257, −1.87891378443402310107045640709, 5.44308394060920118053063766177, 6.78769112702779397449751604521, 8.420391863141385097673542379556, 9.964304215128232789363218962583, 11.31909615551257920893083697409, 13.78513331752079752861621520575, 14.73980320008974528417600094502, 16.20846091400625240615993077031, 17.01605716152957519814599177618, 18.01378780804988666403641028519

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