Properties

Label 2-21-7.3-c8-0-5
Degree $2$
Conductor $21$
Sign $-0.944 - 0.329i$
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  + (15.3 + 26.5i)2-s + (40.5 + 23.3i)3-s + (−341. + 591. i)4-s + (563. − 325. i)5-s + 1.43e3i·6-s + (−1.07e3 + 2.14e3i)7-s − 1.31e4·8-s + (1.09e3 + 1.89e3i)9-s + (1.72e4 + 9.97e3i)10-s + (1.22e4 − 2.12e4i)11-s + (−2.76e4 + 1.59e4i)12-s − 1.95e3i·13-s + (−7.34e4 + 4.47e3i)14-s + 3.04e4·15-s + (−1.13e5 − 1.96e5i)16-s + (5.48e4 + 3.16e4i)17-s + ⋯
L(s)  = 1  + (0.957 + 1.65i)2-s + (0.5 + 0.288i)3-s + (−1.33 + 2.31i)4-s + (0.902 − 0.520i)5-s + 1.10i·6-s + (−0.446 + 0.894i)7-s − 3.19·8-s + (0.166 + 0.288i)9-s + (1.72 + 0.997i)10-s + (0.837 − 1.44i)11-s + (−1.33 + 0.770i)12-s − 0.0683i·13-s + (−1.91 + 0.116i)14-s + 0.601·15-s + (−1.72 − 2.99i)16-s + (0.656 + 0.379i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.944 - 0.329i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.944 - 0.329i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $-0.944 - 0.329i$
Analytic conductor: \(8.55495\)
Root analytic conductor: \(2.92488\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :4),\ -0.944 - 0.329i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.525451 + 3.09796i\)
\(L(\frac12)\) \(\approx\) \(0.525451 + 3.09796i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-40.5 - 23.3i)T \)
7 \( 1 + (1.07e3 - 2.14e3i)T \)
good2 \( 1 + (-15.3 - 26.5i)T + (-128 + 221. i)T^{2} \)
5 \( 1 + (-563. + 325. i)T + (1.95e5 - 3.38e5i)T^{2} \)
11 \( 1 + (-1.22e4 + 2.12e4i)T + (-1.07e8 - 1.85e8i)T^{2} \)
13 \( 1 + 1.95e3iT - 8.15e8T^{2} \)
17 \( 1 + (-5.48e4 - 3.16e4i)T + (3.48e9 + 6.04e9i)T^{2} \)
19 \( 1 + (1.98e4 - 1.14e4i)T + (8.49e9 - 1.47e10i)T^{2} \)
23 \( 1 + (-6.51e4 - 1.12e5i)T + (-3.91e10 + 6.78e10i)T^{2} \)
29 \( 1 - 4.15e5T + 5.00e11T^{2} \)
31 \( 1 + (-9.50e5 - 5.48e5i)T + (4.26e11 + 7.38e11i)T^{2} \)
37 \( 1 + (6.36e5 + 1.10e6i)T + (-1.75e12 + 3.04e12i)T^{2} \)
41 \( 1 + 4.21e6iT - 7.98e12T^{2} \)
43 \( 1 + 1.04e6T + 1.16e13T^{2} \)
47 \( 1 + (1.94e6 - 1.12e6i)T + (1.19e13 - 2.06e13i)T^{2} \)
53 \( 1 + (-1.65e6 + 2.86e6i)T + (-3.11e13 - 5.39e13i)T^{2} \)
59 \( 1 + (1.48e7 + 8.55e6i)T + (7.34e13 + 1.27e14i)T^{2} \)
61 \( 1 + (-2.44e6 + 1.40e6i)T + (9.58e13 - 1.66e14i)T^{2} \)
67 \( 1 + (-1.33e7 + 2.31e7i)T + (-2.03e14 - 3.51e14i)T^{2} \)
71 \( 1 - 1.07e7T + 6.45e14T^{2} \)
73 \( 1 + (2.55e7 + 1.47e7i)T + (4.03e14 + 6.98e14i)T^{2} \)
79 \( 1 + (1.28e7 + 2.23e7i)T + (-7.58e14 + 1.31e15i)T^{2} \)
83 \( 1 - 8.30e7iT - 2.25e15T^{2} \)
89 \( 1 + (-3.82e7 + 2.20e7i)T + (1.96e15 - 3.40e15i)T^{2} \)
97 \( 1 + 3.44e7iT - 7.83e15T^{2} \)
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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

−16.50080855775383351964584823984, −15.56170568676984853602129531811, −14.29078590237407092305109231955, −13.53254426076564100411032550243, −12.30521161923037479516010661128, −9.242281508410003301239495638478, −8.376452875776920374377813698767, −6.32013439030075335173995877740, −5.35060087565109793965804227986, −3.40753390495801416549861755263, 1.35548897443531895879920673947, 2.78425934190061825550457288907, 4.38736658027689976762218467031, 6.54362847528510688314368031474, 9.637216468544292875506582453627, 10.22302537358684795290591588449, 11.92351353306802213992357174457, 13.11042269659700204146893909564, 14.00964334263909275906649123770, 14.82426439646162665567419087438

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