Properties

Label 2-21-21.20-c21-0-14
Degree $2$
Conductor $21$
Sign $-0.441 - 0.897i$
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  − 247. i·2-s + (1.62e4 + 1.00e5i)3-s + 2.03e6·4-s − 3.48e7·5-s + (2.49e7 − 4.00e6i)6-s + (7.14e8 − 2.19e8i)7-s − 1.02e9i·8-s + (−9.93e9 + 3.27e9i)9-s + 8.59e9i·10-s − 9.21e10i·11-s + (3.30e10 + 2.05e11i)12-s + 1.67e11i·13-s + (−5.41e10 − 1.76e11i)14-s + (−5.64e11 − 3.51e12i)15-s + 4.01e12·16-s + 7.92e12·17-s + ⋯
L(s)  = 1  − 0.170i·2-s + (0.158 + 0.987i)3-s + 0.970·4-s − 1.59·5-s + (0.168 − 0.0270i)6-s + (0.955 − 0.293i)7-s − 0.336i·8-s + (−0.949 + 0.312i)9-s + 0.271i·10-s − 1.07i·11-s + (0.153 + 0.958i)12-s + 0.337i·13-s + (−0.0500 − 0.163i)14-s + (−0.252 − 1.57i)15-s + 0.913·16-s + 0.953·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.441 - 0.897i)\, \overline{\Lambda}(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & (-0.441 - 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $-0.441 - 0.897i$
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: \((2,\ 21,\ (\ :21/2),\ -0.441 - 0.897i)\)

Particular Values

\(L(11)\) \(\approx\) \(1.747262467\)
\(L(\frac12)\) \(\approx\) \(1.747262467\)
\(L(\frac{23}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.62e4 - 1.00e5i)T \)
7 \( 1 + (-7.14e8 + 2.19e8i)T \)
good2 \( 1 + 247. iT - 2.09e6T^{2} \)
5 \( 1 + 3.48e7T + 4.76e14T^{2} \)
11 \( 1 + 9.21e10iT - 7.40e21T^{2} \)
13 \( 1 - 1.67e11iT - 2.47e23T^{2} \)
17 \( 1 - 7.92e12T + 6.90e25T^{2} \)
19 \( 1 - 3.32e13iT - 7.14e26T^{2} \)
23 \( 1 - 3.78e14iT - 3.94e28T^{2} \)
29 \( 1 + 2.17e15iT - 5.13e30T^{2} \)
31 \( 1 - 3.73e15iT - 2.08e31T^{2} \)
37 \( 1 + 1.35e16T + 8.55e32T^{2} \)
41 \( 1 + 6.54e16T + 7.38e33T^{2} \)
43 \( 1 - 4.14e15T + 2.00e34T^{2} \)
47 \( 1 + 4.66e17T + 1.30e35T^{2} \)
53 \( 1 - 1.25e18iT - 1.62e36T^{2} \)
59 \( 1 - 5.49e17T + 1.54e37T^{2} \)
61 \( 1 - 7.00e18iT - 3.10e37T^{2} \)
67 \( 1 + 1.30e19T + 2.22e38T^{2} \)
71 \( 1 - 3.38e19iT - 7.52e38T^{2} \)
73 \( 1 - 5.69e19iT - 1.34e39T^{2} \)
79 \( 1 + 5.19e19T + 7.08e39T^{2} \)
83 \( 1 - 1.77e20T + 1.99e40T^{2} \)
89 \( 1 + 4.40e20T + 8.65e40T^{2} \)
97 \( 1 + 5.59e20iT - 5.27e41T^{2} \)
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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

−14.20089064482472664802208761314, −11.77523369799850833563033838272, −11.44389771104258439880515154388, −10.28164448583231030635032703928, −8.327088667395378510636831116013, −7.56355957516678117247991330067, −5.54955515471769180223064571607, −3.93522167795601506393385690716, −3.23942920202680584532024077269, −1.27614629531563527754304230750, 0.44197172261849233523981280200, 1.82941592822276184869134243981, 3.06733062070063073615520881747, 4.89009239027072415675036088042, 6.75530963999685953462671312592, 7.62710226867958786677937377762, 8.383810676206731586933370519111, 10.91355719993217505294290580166, 11.87602781868145588343218846225, 12.51986916396700932968644159754

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