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Properties

Label 2-21-21.20-c21-0-34
Degree $2$
Conductor $21$
Sign $-0.826 - 0.562i$
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  + 1.92e3i·2-s + (7.83e4 + 6.57e4i)3-s − 1.61e6·4-s + 2.77e7·5-s + (−1.26e8 + 1.50e8i)6-s + (7.43e8 − 7.50e7i)7-s + 9.36e8i·8-s + (1.81e9 + 1.03e10i)9-s + 5.34e10i·10-s − 5.61e10i·11-s + (−1.26e11 − 1.05e11i)12-s − 3.86e11i·13-s + (1.44e11 + 1.43e12i)14-s + (2.17e12 + 1.82e12i)15-s − 5.18e12·16-s + 1.33e13·17-s + ⋯
L(s)  = 1  + 1.32i·2-s + (0.766 + 0.642i)3-s − 0.768·4-s + 1.27·5-s + (−0.854 + 1.01i)6-s + (0.994 − 0.100i)7-s + 0.308i·8-s + (0.173 + 0.984i)9-s + 1.69i·10-s − 0.652i·11-s + (−0.588 − 0.493i)12-s − 0.777i·13-s + (0.133 + 1.32i)14-s + (0.973 + 0.816i)15-s − 1.17·16-s + 1.60·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.826 - 0.562i)\, \overline{\Lambda}(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & (-0.826 - 0.562i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $-0.826 - 0.562i$
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: \((2,\ 21,\ (\ :21/2),\ -0.826 - 0.562i)\)

Particular Values

\(L(11)\) \(\approx\) \(4.436443405\)
\(L(\frac12)\) \(\approx\) \(4.436443405\)
\(L(\frac{23}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-7.83e4 - 6.57e4i)T \)
7 \( 1 + (-7.43e8 + 7.50e7i)T \)
good2 \( 1 - 1.92e3iT - 2.09e6T^{2} \)
5 \( 1 - 2.77e7T + 4.76e14T^{2} \)
11 \( 1 + 5.61e10iT - 7.40e21T^{2} \)
13 \( 1 + 3.86e11iT - 2.47e23T^{2} \)
17 \( 1 - 1.33e13T + 6.90e25T^{2} \)
19 \( 1 + 3.04e13iT - 7.14e26T^{2} \)
23 \( 1 - 2.80e14iT - 3.94e28T^{2} \)
29 \( 1 - 2.28e15iT - 5.13e30T^{2} \)
31 \( 1 - 3.90e15iT - 2.08e31T^{2} \)
37 \( 1 - 3.61e16T + 8.55e32T^{2} \)
41 \( 1 - 7.43e16T + 7.38e33T^{2} \)
43 \( 1 + 1.37e17T + 2.00e34T^{2} \)
47 \( 1 + 1.27e17T + 1.30e35T^{2} \)
53 \( 1 + 1.55e18iT - 1.62e36T^{2} \)
59 \( 1 + 6.38e18T + 1.54e37T^{2} \)
61 \( 1 - 1.92e18iT - 3.10e37T^{2} \)
67 \( 1 - 2.86e18T + 2.22e38T^{2} \)
71 \( 1 + 2.30e19iT - 7.52e38T^{2} \)
73 \( 1 - 1.86e19iT - 1.34e39T^{2} \)
79 \( 1 + 1.57e20T + 7.08e39T^{2} \)
83 \( 1 - 1.76e20T + 1.99e40T^{2} \)
89 \( 1 + 3.24e20T + 8.65e40T^{2} \)
97 \( 1 + 2.37e20iT - 5.27e41T^{2} \)
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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

−14.22523890458530852359010355381, −13.44515715330512122802897759506, −11.01454606197142674916427711280, −9.646413857556784688761190933077, −8.462233745890063762798624964927, −7.46771998437267465030625581894, −5.69184203043985239044485643561, −5.02443029950908040983886663893, −2.99957388923556746561127367736, −1.49967046990245569681273843664, 1.11836811726368479469739094773, 1.83793937690020937402946676194, 2.60624360850438574235854826479, 4.23025222655479909149767620685, 6.13840568614792929213324724668, 7.82515082542635487693799918309, 9.379667404642913168865731471226, 10.18100176313355168094137410560, 11.80577977335197722903845025054, 12.70391322074908179589855654881

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