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Properties

Label 2-21-21.20-c21-0-51
Degree $2$
Conductor $21$
Sign $-0.348 - 0.937i$
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  − 1.07e3i·2-s + (6.48e4 − 7.90e4i)3-s + 9.35e5·4-s − 7.14e6·5-s + (−8.52e7 − 6.98e7i)6-s + (−3.76e8 + 6.45e8i)7-s − 3.26e9i·8-s + (−2.04e9 − 1.02e10i)9-s + 7.69e9i·10-s − 6.21e8i·11-s + (6.06e10 − 7.39e10i)12-s − 3.37e11i·13-s + (6.95e11 + 4.05e11i)14-s + (−4.63e11 + 5.64e11i)15-s − 1.56e12·16-s − 6.87e12·17-s + ⋯
L(s)  = 1  − 0.744i·2-s + (0.634 − 0.773i)3-s + 0.446·4-s − 0.326·5-s + (−0.575 − 0.471i)6-s + (−0.503 + 0.864i)7-s − 1.07i·8-s + (−0.195 − 0.980i)9-s + 0.243i·10-s − 0.00722i·11-s + (0.282 − 0.344i)12-s − 0.679i·13-s + (0.643 + 0.374i)14-s + (−0.207 + 0.252i)15-s − 0.354·16-s − 0.826·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.348 - 0.937i)\, \overline{\Lambda}(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & (-0.348 - 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $-0.348 - 0.937i$
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: \((2,\ 21,\ (\ :21/2),\ -0.348 - 0.937i)\)

Particular Values

\(L(11)\) \(\approx\) \(0.6682148745\)
\(L(\frac12)\) \(\approx\) \(0.6682148745\)
\(L(\frac{23}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-6.48e4 + 7.90e4i)T \)
7 \( 1 + (3.76e8 - 6.45e8i)T \)
good2 \( 1 + 1.07e3iT - 2.09e6T^{2} \)
5 \( 1 + 7.14e6T + 4.76e14T^{2} \)
11 \( 1 + 6.21e8iT - 7.40e21T^{2} \)
13 \( 1 + 3.37e11iT - 2.47e23T^{2} \)
17 \( 1 + 6.87e12T + 6.90e25T^{2} \)
19 \( 1 - 1.17e13iT - 7.14e26T^{2} \)
23 \( 1 - 8.43e13iT - 3.94e28T^{2} \)
29 \( 1 + 6.05e14iT - 5.13e30T^{2} \)
31 \( 1 - 1.22e15iT - 2.08e31T^{2} \)
37 \( 1 + 3.62e16T + 8.55e32T^{2} \)
41 \( 1 + 1.58e16T + 7.38e33T^{2} \)
43 \( 1 + 2.84e16T + 2.00e34T^{2} \)
47 \( 1 + 2.36e17T + 1.30e35T^{2} \)
53 \( 1 - 6.88e17iT - 1.62e36T^{2} \)
59 \( 1 - 4.19e18T + 1.54e37T^{2} \)
61 \( 1 - 5.56e18iT - 3.10e37T^{2} \)
67 \( 1 + 1.57e19T + 2.22e38T^{2} \)
71 \( 1 - 7.43e18iT - 7.52e38T^{2} \)
73 \( 1 - 4.27e19iT - 1.34e39T^{2} \)
79 \( 1 + 1.56e20T + 7.08e39T^{2} \)
83 \( 1 - 1.29e20T + 1.99e40T^{2} \)
89 \( 1 - 4.94e20T + 8.65e40T^{2} \)
97 \( 1 + 5.62e19iT - 5.27e41T^{2} \)
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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

−12.40706914071846141200675144835, −11.55756312913554015645253973735, −9.948680488872053643621000149443, −8.597408073276414241898045182015, −7.22134017023141443607730894244, −5.99126889057476629882652769034, −3.61556492105731223728271523577, −2.63600895987007129716558688792, −1.61279848603810972864978467798, −0.13438898059480038116642328013, 2.07805143246145755101870216903, 3.52753489121496517031574648849, 4.78234145951491706174585975212, 6.53449130347288021715104320906, 7.62645406644643759085362754178, 8.899838129841119954973347139043, 10.34656270682224302394215389823, 11.47263367648400391691952888095, 13.45372590429366072813378225542, 14.50746767837224310035792052192

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