Properties

Label 2-21-21.20-c21-0-46
Degree $2$
Conductor $21$
Sign $-0.460 + 0.887i$
Analytic cond. $58.6902$
Root an. cond. $7.66095$
Motivic weight $21$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 687. i·2-s + (1.00e5 − 1.89e4i)3-s + 1.62e6·4-s − 6.28e6·5-s + (−1.30e7 − 6.91e7i)6-s + (4.61e8 − 5.87e8i)7-s − 2.55e9i·8-s + (9.74e9 − 3.80e9i)9-s + 4.32e9i·10-s − 6.92e10i·11-s + (1.63e11 − 3.07e10i)12-s + 2.42e11i·13-s + (−4.04e11 − 3.17e11i)14-s + (−6.31e11 + 1.19e11i)15-s + 1.64e12·16-s − 6.97e12·17-s + ⋯
L(s)  = 1  − 0.474i·2-s + (0.982 − 0.185i)3-s + 0.774·4-s − 0.287·5-s + (−0.0879 − 0.466i)6-s + (0.617 − 0.786i)7-s − 0.842i·8-s + (0.931 − 0.364i)9-s + 0.136i·10-s − 0.804i·11-s + (0.761 − 0.143i)12-s + 0.486i·13-s + (−0.373 − 0.293i)14-s + (−0.282 + 0.0533i)15-s + 0.374·16-s − 0.838·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.460 + 0.887i)\, \overline{\Lambda}(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & (-0.460 + 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $-0.460 + 0.887i$
Analytic conductor: \(58.6902\)
Root analytic conductor: \(7.66095\)
Motivic weight: \(21\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :21/2),\ -0.460 + 0.887i)\)

Particular Values

\(L(11)\) \(\approx\) \(3.801691670\)
\(L(\frac12)\) \(\approx\) \(3.801691670\)
\(L(\frac{23}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.00e5 + 1.89e4i)T \)
7 \( 1 + (-4.61e8 + 5.87e8i)T \)
good2 \( 1 + 687. iT - 2.09e6T^{2} \)
5 \( 1 + 6.28e6T + 4.76e14T^{2} \)
11 \( 1 + 6.92e10iT - 7.40e21T^{2} \)
13 \( 1 - 2.42e11iT - 2.47e23T^{2} \)
17 \( 1 + 6.97e12T + 6.90e25T^{2} \)
19 \( 1 + 3.12e13iT - 7.14e26T^{2} \)
23 \( 1 + 5.39e13iT - 3.94e28T^{2} \)
29 \( 1 - 2.80e15iT - 5.13e30T^{2} \)
31 \( 1 - 2.80e15iT - 2.08e31T^{2} \)
37 \( 1 - 3.19e16T + 8.55e32T^{2} \)
41 \( 1 - 1.94e16T + 7.38e33T^{2} \)
43 \( 1 - 9.38e16T + 2.00e34T^{2} \)
47 \( 1 - 3.19e16T + 1.30e35T^{2} \)
53 \( 1 + 1.47e18iT - 1.62e36T^{2} \)
59 \( 1 + 5.35e18T + 1.54e37T^{2} \)
61 \( 1 - 9.25e18iT - 3.10e37T^{2} \)
67 \( 1 - 8.77e18T + 2.22e38T^{2} \)
71 \( 1 + 4.46e19iT - 7.52e38T^{2} \)
73 \( 1 + 4.34e19iT - 1.34e39T^{2} \)
79 \( 1 - 6.43e18T + 7.08e39T^{2} \)
83 \( 1 + 2.21e20T + 1.99e40T^{2} \)
89 \( 1 + 6.58e19T + 8.65e40T^{2} \)
97 \( 1 - 4.64e20iT - 5.27e41T^{2} \)
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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

−13.05362331932650200850624780094, −11.54192229130089585140981207725, −10.60301188917178813514578624872, −9.015046749838678289862319598985, −7.67141010926132137178789407277, −6.67092450983964325321048547564, −4.32735183106917979206405347064, −3.14083044678403226985210162412, −1.93744942030031151154076949096, −0.788896054355541019855023217248, 1.75619376627039539980881913141, 2.60180704965426097858079436955, 4.27391834869052166392451016366, 5.88502181177408097157873999120, 7.53168331608007731618801184455, 8.228644282397628108812763494427, 9.744375912775645174113720543176, 11.28381269597124853464043030172, 12.55366178958002189946992462008, 14.21275202912165010471822473005

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