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Properties

Label 2-21-7.6-c8-0-5
Degree $2$
Conductor $21$
Sign $0.610 + 0.792i$
Analytic cond. $8.55495$
Root an. cond. $2.92488$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.60·2-s + 46.7i·3-s − 249.·4-s − 225. i·5-s − 121. i·6-s + (1.90e3 − 1.46e3i)7-s + 1.31e3·8-s − 2.18e3·9-s + 587. i·10-s + 1.70e4·11-s − 1.16e4i·12-s − 3.90e4i·13-s + (−4.95e3 + 3.81e3i)14-s + 1.05e4·15-s + 6.03e4·16-s − 1.53e5i·17-s + ⋯
L(s)  = 1  − 0.162·2-s + 0.577i·3-s − 0.973·4-s − 0.360i·5-s − 0.0939i·6-s + (0.792 − 0.610i)7-s + 0.321·8-s − 0.333·9-s + 0.0587i·10-s + 1.16·11-s − 0.562i·12-s − 1.36i·13-s + (−0.129 + 0.0993i)14-s + 0.208·15-s + 0.921·16-s − 1.84i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.610 + 0.792i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.610 + 0.792i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $0.610 + 0.792i$
Analytic conductor: \(8.55495\)
Root analytic conductor: \(2.92488\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :4),\ 0.610 + 0.792i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.10402 - 0.543363i\)
\(L(\frac12)\) \(\approx\) \(1.10402 - 0.543363i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 46.7iT \)
7 \( 1 + (-1.90e3 + 1.46e3i)T \)
good2 \( 1 + 2.60T + 256T^{2} \)
5 \( 1 + 225. iT - 3.90e5T^{2} \)
11 \( 1 - 1.70e4T + 2.14e8T^{2} \)
13 \( 1 + 3.90e4iT - 8.15e8T^{2} \)
17 \( 1 + 1.53e5iT - 6.97e9T^{2} \)
19 \( 1 - 1.24e5iT - 1.69e10T^{2} \)
23 \( 1 + 4.22e5T + 7.83e10T^{2} \)
29 \( 1 + 2.19e5T + 5.00e11T^{2} \)
31 \( 1 - 1.37e5iT - 8.52e11T^{2} \)
37 \( 1 - 1.02e6T + 3.51e12T^{2} \)
41 \( 1 + 4.43e6iT - 7.98e12T^{2} \)
43 \( 1 - 2.57e6T + 1.16e13T^{2} \)
47 \( 1 + 2.73e6iT - 2.38e13T^{2} \)
53 \( 1 + 4.37e6T + 6.22e13T^{2} \)
59 \( 1 - 3.69e6iT - 1.46e14T^{2} \)
61 \( 1 - 1.38e7iT - 1.91e14T^{2} \)
67 \( 1 + 8.84e6T + 4.06e14T^{2} \)
71 \( 1 - 2.56e7T + 6.45e14T^{2} \)
73 \( 1 + 1.19e7iT - 8.06e14T^{2} \)
79 \( 1 - 4.30e7T + 1.51e15T^{2} \)
83 \( 1 + 2.79e7iT - 2.25e15T^{2} \)
89 \( 1 - 8.40e7iT - 3.93e15T^{2} \)
97 \( 1 + 5.93e7iT - 7.83e15T^{2} \)
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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

−16.32111463233591710893893538526, −14.60334893516516821535544453918, −13.77377396395378770638098172002, −12.08959173136812777784318104346, −10.42389018457307692416537398313, −9.200647034582824980581610177452, −7.86823306820854851579458045363, −5.26443225953709792054543322236, −3.94471723092793261156527717084, −0.75357873145781763465004503884, 1.61351261651767414637123178455, 4.27679332285526507026540158292, 6.29888921651769138909064818249, 8.195821435740912476361377757686, 9.305836687467385790991239605252, 11.25194215883826644368574453395, 12.54186729204181744375569251464, 14.05510780174292657715961738743, 14.76773706065782308132853130452, 16.89877237990091210544189254387

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