Properties

Label 2-21-21.2-c6-0-3
Degree $2$
Conductor $21$
Sign $0.306 + 0.951i$
Analytic cond. $4.83113$
Root an. cond. $2.19798$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−11.1 − 6.41i)2-s + (−14.0 + 23.0i)3-s + (50.2 + 87.1i)4-s + (38.4 + 22.2i)5-s + (303. − 166. i)6-s + (−244. − 240. i)7-s − 469. i·8-s + (−334. − 647. i)9-s + (−284. − 493. i)10-s + (1.97e3 − 1.14e3i)11-s + (−2.71e3 − 63.9i)12-s + 1.63e3·13-s + (1.17e3 + 4.23e3i)14-s + (−1.05e3 + 574. i)15-s + (207. − 360. i)16-s + (−565. + 326. i)17-s + ⋯
L(s)  = 1  + (−1.38 − 0.801i)2-s + (−0.520 + 0.854i)3-s + (0.785 + 1.36i)4-s + (0.307 + 0.177i)5-s + (1.40 − 0.768i)6-s + (−0.713 − 0.700i)7-s − 0.916i·8-s + (−0.458 − 0.888i)9-s + (−0.284 − 0.493i)10-s + (1.48 − 0.857i)11-s + (−1.57 − 0.0370i)12-s + 0.743·13-s + (0.429 + 1.54i)14-s + (−0.311 + 0.170i)15-s + (0.0507 − 0.0879i)16-s + (−0.115 + 0.0664i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.306 + 0.951i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.306 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $0.306 + 0.951i$
Analytic conductor: \(4.83113\)
Root analytic conductor: \(2.19798\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :3),\ 0.306 + 0.951i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.478973 - 0.349072i\)
\(L(\frac12)\) \(\approx\) \(0.478973 - 0.349072i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (14.0 - 23.0i)T \)
7 \( 1 + (244. + 240. i)T \)
good2 \( 1 + (11.1 + 6.41i)T + (32 + 55.4i)T^{2} \)
5 \( 1 + (-38.4 - 22.2i)T + (7.81e3 + 1.35e4i)T^{2} \)
11 \( 1 + (-1.97e3 + 1.14e3i)T + (8.85e5 - 1.53e6i)T^{2} \)
13 \( 1 - 1.63e3T + 4.82e6T^{2} \)
17 \( 1 + (565. - 326. i)T + (1.20e7 - 2.09e7i)T^{2} \)
19 \( 1 + (-1.43e3 + 2.48e3i)T + (-2.35e7 - 4.07e7i)T^{2} \)
23 \( 1 + (-5.70e3 - 3.29e3i)T + (7.40e7 + 1.28e8i)T^{2} \)
29 \( 1 + 3.19e4iT - 5.94e8T^{2} \)
31 \( 1 + (-2.34e4 - 4.06e4i)T + (-4.43e8 + 7.68e8i)T^{2} \)
37 \( 1 + (-1.28e4 + 2.22e4i)T + (-1.28e9 - 2.22e9i)T^{2} \)
41 \( 1 + 9.09e4iT - 4.75e9T^{2} \)
43 \( 1 + 8.77e4T + 6.32e9T^{2} \)
47 \( 1 + (-1.40e4 - 8.12e3i)T + (5.38e9 + 9.33e9i)T^{2} \)
53 \( 1 + (-5.49e4 + 3.17e4i)T + (1.10e10 - 1.91e10i)T^{2} \)
59 \( 1 + (5.73e4 - 3.31e4i)T + (2.10e10 - 3.65e10i)T^{2} \)
61 \( 1 + (-2.26e5 + 3.92e5i)T + (-2.57e10 - 4.46e10i)T^{2} \)
67 \( 1 + (-2.23e4 - 3.87e4i)T + (-4.52e10 + 7.83e10i)T^{2} \)
71 \( 1 + 1.33e5iT - 1.28e11T^{2} \)
73 \( 1 + (2.59e5 + 4.49e5i)T + (-7.56e10 + 1.31e11i)T^{2} \)
79 \( 1 + (7.38e4 - 1.27e5i)T + (-1.21e11 - 2.10e11i)T^{2} \)
83 \( 1 - 2.81e5iT - 3.26e11T^{2} \)
89 \( 1 + (-4.69e5 - 2.71e5i)T + (2.48e11 + 4.30e11i)T^{2} \)
97 \( 1 - 8.95e4T + 8.32e11T^{2} \)
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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

−16.93645967590146744694050754800, −15.95054776428716972055779621969, −13.95085215455330499168658957076, −11.84839903814584014882543254496, −10.84126965641598916967298422045, −9.804637407353480545905202009510, −8.784146687867840451423438590822, −6.44532004197380059715704049037, −3.55970826715938248186539746886, −0.72807799989705585206589816515, 1.36050624872566021364316224502, 6.03476883729969580697513278244, 6.99047306310285389375177778975, 8.649087040599358057492542455294, 9.784120311162199330505210301225, 11.64508624518045331172696135981, 13.07113665770811319733658852859, 14.93786509509914514871475921739, 16.38302802687969140590795157182, 17.16710459672251386841398295895

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