Properties

Label 2-21-21.20-c21-0-47
Degree $2$
Conductor $21$
Sign $0.271 + 0.962i$
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  + 1.48e3i·2-s + (5.69e4 − 8.49e4i)3-s − 1.10e5·4-s + 3.09e6·5-s + (1.26e8 + 8.45e7i)6-s + (4.84e8 − 5.68e8i)7-s + 2.95e9i·8-s + (−3.98e9 − 9.67e9i)9-s + 4.59e9i·10-s − 1.18e11i·11-s + (−6.28e9 + 9.38e9i)12-s − 4.83e11i·13-s + (8.45e11 + 7.20e11i)14-s + (1.76e11 − 2.62e11i)15-s − 4.61e12·16-s − 9.41e10·17-s + ⋯
L(s)  = 1  + 1.02i·2-s + (0.556 − 0.830i)3-s − 0.0526·4-s + 0.141·5-s + (0.852 + 0.570i)6-s + (0.648 − 0.761i)7-s + 0.971i·8-s + (−0.380 − 0.924i)9-s + 0.145i·10-s − 1.37i·11-s + (−0.0292 + 0.0437i)12-s − 0.973i·13-s + (0.780 + 0.665i)14-s + (0.0788 − 0.117i)15-s − 1.04·16-s − 0.0113·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.271 + 0.962i)\, \overline{\Lambda}(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & (0.271 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $0.271 + 0.962i$
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: \((2,\ 21,\ (\ :21/2),\ 0.271 + 0.962i)\)

Particular Values

\(L(11)\) \(\approx\) \(2.459475062\)
\(L(\frac12)\) \(\approx\) \(2.459475062\)
\(L(\frac{23}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-5.69e4 + 8.49e4i)T \)
7 \( 1 + (-4.84e8 + 5.68e8i)T \)
good2 \( 1 - 1.48e3iT - 2.09e6T^{2} \)
5 \( 1 - 3.09e6T + 4.76e14T^{2} \)
11 \( 1 + 1.18e11iT - 7.40e21T^{2} \)
13 \( 1 + 4.83e11iT - 2.47e23T^{2} \)
17 \( 1 + 9.41e10T + 6.90e25T^{2} \)
19 \( 1 - 3.71e13iT - 7.14e26T^{2} \)
23 \( 1 + 1.88e14iT - 3.94e28T^{2} \)
29 \( 1 + 3.79e15iT - 5.13e30T^{2} \)
31 \( 1 - 7.93e15iT - 2.08e31T^{2} \)
37 \( 1 + 4.02e16T + 8.55e32T^{2} \)
41 \( 1 + 1.25e17T + 7.38e33T^{2} \)
43 \( 1 + 3.23e15T + 2.00e34T^{2} \)
47 \( 1 - 4.91e17T + 1.30e35T^{2} \)
53 \( 1 + 5.38e15iT - 1.62e36T^{2} \)
59 \( 1 - 9.12e17T + 1.54e37T^{2} \)
61 \( 1 + 6.80e18iT - 3.10e37T^{2} \)
67 \( 1 - 2.32e19T + 2.22e38T^{2} \)
71 \( 1 + 2.32e19iT - 7.52e38T^{2} \)
73 \( 1 + 1.56e19iT - 1.34e39T^{2} \)
79 \( 1 - 8.59e19T + 7.08e39T^{2} \)
83 \( 1 - 1.01e20T + 1.99e40T^{2} \)
89 \( 1 - 5.73e19T + 8.65e40T^{2} \)
97 \( 1 + 6.72e20iT - 5.27e41T^{2} \)
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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

−13.70171239763206894101888065713, −12.00402073522520823181114066320, −10.56893072195564397490921030754, −8.390291802732193662154037558661, −7.912740668802756196966118240159, −6.58737958428107779400617273005, −5.52772924376407668803315078766, −3.44975559403678080357398432969, −1.89136247253188371355344921751, −0.51699920408795950860529985438, 1.75974914511848438335889971387, 2.41122755570902637577677652122, 3.89653251011826121977584628974, 5.06729114228070146445984780581, 7.18361076870512790482463587023, 8.992894954984536296412504324259, 9.815808450497007387521518060808, 11.12414443458495576358379407080, 12.03910240486646985981383322044, 13.54569409094803502914549544039

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