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Properties

Label 2-21-7.4-c1-0-0
Degree $2$
Conductor $21$
Sign $0.605 + 0.795i$
Analytic cond. $0.167685$
Root an. cond. $0.409494$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 − 1.73i)2-s + (−0.5 + 0.866i)3-s + (−0.999 + 1.73i)4-s + (1 + 1.73i)5-s + 1.99·6-s + (−2.5 − 0.866i)7-s + (−0.499 − 0.866i)9-s + (1.99 − 3.46i)10-s + (1 − 1.73i)11-s + (−1 − 1.73i)12-s + 13-s + (1.00 + 5.19i)14-s − 1.99·15-s + (1.99 + 3.46i)16-s + (−0.999 + 1.73i)18-s + (−0.5 − 0.866i)19-s + ⋯
L(s)  = 1  + (−0.707 − 1.22i)2-s + (−0.288 + 0.499i)3-s + (−0.499 + 0.866i)4-s + (0.447 + 0.774i)5-s + 0.816·6-s + (−0.944 − 0.327i)7-s + (−0.166 − 0.288i)9-s + (0.632 − 1.09i)10-s + (0.301 − 0.522i)11-s + (−0.288 − 0.499i)12-s + 0.277·13-s + (0.267 + 1.38i)14-s − 0.516·15-s + (0.499 + 0.866i)16-s + (−0.235 + 0.408i)18-s + (−0.114 − 0.198i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $0.605 + 0.795i$
Analytic conductor: \(0.167685\)
Root analytic conductor: \(0.409494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :1/2),\ 0.605 + 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.389127 - 0.192901i\)
\(L(\frac12)\) \(\approx\) \(0.389127 - 0.192901i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (2.5 + 0.866i)T \)
good2 \( 1 + (1 + 1.73i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-1 - 1.73i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1 + 1.73i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - T + 13T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 4T + 29T^{2} \)
31 \( 1 + (4.5 - 7.79i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.5 + 2.59i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 10T + 41T^{2} \)
43 \( 1 - 5T + 43T^{2} \)
47 \( 1 + (-3 - 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (6 - 10.3i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6 + 10.3i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5 + 8.66i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.5 + 4.33i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + (-1.5 + 2.59i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 6T + 83T^{2} \)
89 \( 1 + (8 + 13.8i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 6T + 97T^{2} \)
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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

−18.35162599690320300427413645604, −17.20622136474437902816441882031, −15.79761123923961004510355109668, −14.09592991053196328885893654920, −12.46535281282447583047007665638, −10.97701824034660861287431573674, −10.22619556082021660311570594860, −9.036766158069173391425947412212, −6.38832905030081022881130156037, −3.25829500936861479080741427724, 5.69031773444673653876183633556, 6.94429713931444545466421079467, 8.601280514767607011404874617841, 9.745945150935527986080392896027, 12.15250100734095017521574237444, 13.34323482627396490642499025152, 15.07414260816926352661587808986, 16.32959379130278819697921305148, 17.04031378858136437073408927109, 18.12129056260894093908436548569

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