Properties

Label 2-21-21.20-c21-0-17
Degree $2$
Conductor $21$
Sign $-0.699 - 0.714i$
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  + 1.70e3i·2-s + (−6.58e4 + 7.82e4i)3-s − 8.08e5·4-s − 2.49e7·5-s + (−1.33e8 − 1.12e8i)6-s + (7.21e7 − 7.43e8i)7-s + 2.19e9i·8-s + (−1.80e9 − 1.03e10i)9-s − 4.25e10i·10-s + 5.38e10i·11-s + (5.32e10 − 6.33e10i)12-s − 8.15e11i·13-s + (1.26e12 + 1.22e11i)14-s + (1.64e12 − 1.95e12i)15-s − 5.43e12·16-s + 7.14e12·17-s + ⋯
L(s)  = 1  + 1.17i·2-s + (−0.643 + 0.765i)3-s − 0.385·4-s − 1.14·5-s + (−0.901 − 0.757i)6-s + (0.0964 − 0.995i)7-s + 0.723i·8-s + (−0.172 − 0.985i)9-s − 1.34i·10-s + 0.625i·11-s + (0.248 − 0.295i)12-s − 1.64i·13-s + (1.17 + 0.113i)14-s + (0.736 − 0.875i)15-s − 1.23·16-s + 0.859·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.699 - 0.714i)\, \overline{\Lambda}(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & (-0.699 - 0.714i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $-0.699 - 0.714i$
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: \((2,\ 21,\ (\ :21/2),\ -0.699 - 0.714i)\)

Particular Values

\(L(11)\) \(\approx\) \(1.098281465\)
\(L(\frac12)\) \(\approx\) \(1.098281465\)
\(L(\frac{23}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (6.58e4 - 7.82e4i)T \)
7 \( 1 + (-7.21e7 + 7.43e8i)T \)
good2 \( 1 - 1.70e3iT - 2.09e6T^{2} \)
5 \( 1 + 2.49e7T + 4.76e14T^{2} \)
11 \( 1 - 5.38e10iT - 7.40e21T^{2} \)
13 \( 1 + 8.15e11iT - 2.47e23T^{2} \)
17 \( 1 - 7.14e12T + 6.90e25T^{2} \)
19 \( 1 - 2.64e13iT - 7.14e26T^{2} \)
23 \( 1 + 9.27e13iT - 3.94e28T^{2} \)
29 \( 1 - 2.96e15iT - 5.13e30T^{2} \)
31 \( 1 + 4.95e15iT - 2.08e31T^{2} \)
37 \( 1 - 3.48e16T + 8.55e32T^{2} \)
41 \( 1 - 8.41e15T + 7.38e33T^{2} \)
43 \( 1 + 7.53e15T + 2.00e34T^{2} \)
47 \( 1 - 2.39e17T + 1.30e35T^{2} \)
53 \( 1 + 1.77e18iT - 1.62e36T^{2} \)
59 \( 1 + 6.53e18T + 1.54e37T^{2} \)
61 \( 1 + 4.21e17iT - 3.10e37T^{2} \)
67 \( 1 + 1.30e19T + 2.22e38T^{2} \)
71 \( 1 - 3.56e19iT - 7.52e38T^{2} \)
73 \( 1 - 3.10e19iT - 1.34e39T^{2} \)
79 \( 1 - 1.16e20T + 7.08e39T^{2} \)
83 \( 1 + 1.17e19T + 1.99e40T^{2} \)
89 \( 1 + 8.40e19T + 8.65e40T^{2} \)
97 \( 1 - 4.28e20iT - 5.27e41T^{2} \)
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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

−14.55689955556696891901841305698, −12.50025678434264784550716553750, −11.17332675382040699012093654243, −10.10763238554854618500384645378, −8.100669407413510953781596688031, −7.33618466158666036038813264385, −5.85230373923146669900803049723, −4.64992299392273498742932911337, −3.46945418460901322011445881815, −0.68473709045379592832470183083, 0.50489427998603740506658908877, 1.72422864269485666040233591318, 2.94551291195689510794696261938, 4.46530128587114650773637317472, 6.23783317308687379224952880618, 7.60769604634904909029524026728, 9.145990078149925233982086893682, 11.02083342512929024122321197895, 11.75682681806181927884623543506, 12.22983721374074146124866204089

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