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Properties

Label 2-21-21.17-c3-0-0
Degree $2$
Conductor $21$
Sign $-0.473 - 0.880i$
Analytic cond. $1.23904$
Root an. cond. $1.11312$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.65 + 0.954i)2-s + (−3.47 + 3.86i)3-s + (−2.17 + 3.77i)4-s + (0.623 + 1.08i)5-s + (2.05 − 9.70i)6-s + (10.0 + 15.5i)7-s − 23.5i·8-s + (−2.84 − 26.8i)9-s + (−2.06 − 1.19i)10-s + (35.2 + 20.3i)11-s + (−7.00 − 21.5i)12-s + 19.5i·13-s + (−31.4 − 16.0i)14-s + (−6.34 − 1.34i)15-s + (5.08 + 8.80i)16-s + (−52.3 + 90.6i)17-s + ⋯
L(s)  = 1  + (−0.584 + 0.337i)2-s + (−0.668 + 0.743i)3-s + (−0.272 + 0.471i)4-s + (0.0557 + 0.0966i)5-s + (0.140 − 0.660i)6-s + (0.544 + 0.838i)7-s − 1.04i·8-s + (−0.105 − 0.994i)9-s + (−0.0652 − 0.0376i)10-s + (0.965 + 0.557i)11-s + (−0.168 − 0.517i)12-s + 0.418i·13-s + (−0.601 − 0.306i)14-s + (−0.109 − 0.0231i)15-s + (0.0794 + 0.137i)16-s + (−0.746 + 1.29i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.473 - 0.880i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.473 - 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $-0.473 - 0.880i$
Analytic conductor: \(1.23904\)
Root analytic conductor: \(1.11312\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :3/2),\ -0.473 - 0.880i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.333472 + 0.557856i\)
\(L(\frac12)\) \(\approx\) \(0.333472 + 0.557856i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (3.47 - 3.86i)T \)
7 \( 1 + (-10.0 - 15.5i)T \)
good2 \( 1 + (1.65 - 0.954i)T + (4 - 6.92i)T^{2} \)
5 \( 1 + (-0.623 - 1.08i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-35.2 - 20.3i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 19.5iT - 2.19e3T^{2} \)
17 \( 1 + (52.3 - 90.6i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-35.0 + 20.2i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-69.6 + 40.2i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 211. iT - 2.43e4T^{2} \)
31 \( 1 + (86.6 + 50.0i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-94.9 - 164. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 186.T + 6.89e4T^{2} \)
43 \( 1 - 158.T + 7.95e4T^{2} \)
47 \( 1 + (-179. - 310. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (366. + 211. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-312. + 541. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-699. + 403. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (149. - 258. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 455. iT - 3.57e5T^{2} \)
73 \( 1 + (434. + 250. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-30.9 - 53.6i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 73.1T + 5.71e5T^{2} \)
89 \( 1 + (-57.3 - 99.3i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 1.41e3iT - 9.12e5T^{2} \)
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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

−17.66827196137642164987904997909, −17.10911818888887647648049021537, −15.78483766461018184557735671976, −14.70556678156070905144050214077, −12.59042027457219781827028116359, −11.40296697544931254125047105160, −9.670094130622604929660241865842, −8.572337034394272235163533780154, −6.48798872909125731404408245587, −4.35196236005114589951852683636, 1.07429463353818485527786137358, 5.23094047433692741916386119015, 7.18272414769056445025437484715, 8.962404891620547372211015057556, 10.71498971576783047374582640424, 11.56358042536175209674486540391, 13.40539165902188267968038589392, 14.40787644068455458982485843518, 16.54243615125508580197779135492, 17.56169231462131925902885993725

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