Properties

Label 2-21-21.11-c6-0-7
Degree $2$
Conductor $21$
Sign $0.859 - 0.510i$
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  + (−1.07 + 0.621i)2-s + (26.6 − 4.11i)3-s + (−31.2 + 54.0i)4-s + (93.7 − 54.1i)5-s + (−26.1 + 21.0i)6-s + (147. + 309. i)7-s − 157. i·8-s + (695. − 219. i)9-s + (−67.2 + 116. i)10-s + (1.89e3 + 1.09e3i)11-s + (−610. + 1.57e3i)12-s − 1.13e3·13-s + (−351. − 241. i)14-s + (2.27e3 − 1.83e3i)15-s + (−1.90e3 − 3.29e3i)16-s + (−6.94e3 − 4.01e3i)17-s + ⋯
L(s)  = 1  + (−0.134 + 0.0776i)2-s + (0.988 − 0.152i)3-s + (−0.487 + 0.845i)4-s + (0.749 − 0.433i)5-s + (−0.121 + 0.0972i)6-s + (0.431 + 0.902i)7-s − 0.306i·8-s + (0.953 − 0.301i)9-s + (−0.0672 + 0.116i)10-s + (1.42 + 0.822i)11-s + (−0.353 + 0.909i)12-s − 0.518·13-s + (−0.128 − 0.0878i)14-s + (0.675 − 0.542i)15-s + (−0.464 − 0.803i)16-s + (−1.41 − 0.816i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 - 0.510i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.859 - 0.510i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $0.859 - 0.510i$
Analytic conductor: \(4.83113\)
Root analytic conductor: \(2.19798\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :3),\ 0.859 - 0.510i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.91597 + 0.525781i\)
\(L(\frac12)\) \(\approx\) \(1.91597 + 0.525781i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-26.6 + 4.11i)T \)
7 \( 1 + (-147. - 309. i)T \)
good2 \( 1 + (1.07 - 0.621i)T + (32 - 55.4i)T^{2} \)
5 \( 1 + (-93.7 + 54.1i)T + (7.81e3 - 1.35e4i)T^{2} \)
11 \( 1 + (-1.89e3 - 1.09e3i)T + (8.85e5 + 1.53e6i)T^{2} \)
13 \( 1 + 1.13e3T + 4.82e6T^{2} \)
17 \( 1 + (6.94e3 + 4.01e3i)T + (1.20e7 + 2.09e7i)T^{2} \)
19 \( 1 + (3.23e3 + 5.60e3i)T + (-2.35e7 + 4.07e7i)T^{2} \)
23 \( 1 + (-1.11e3 + 642. i)T + (7.40e7 - 1.28e8i)T^{2} \)
29 \( 1 + 1.42e4iT - 5.94e8T^{2} \)
31 \( 1 + (8.12e3 - 1.40e4i)T + (-4.43e8 - 7.68e8i)T^{2} \)
37 \( 1 + (1.66e4 + 2.88e4i)T + (-1.28e9 + 2.22e9i)T^{2} \)
41 \( 1 + 9.86e4iT - 4.75e9T^{2} \)
43 \( 1 - 6.21e4T + 6.32e9T^{2} \)
47 \( 1 + (5.03e4 - 2.90e4i)T + (5.38e9 - 9.33e9i)T^{2} \)
53 \( 1 + (-1.57e5 - 9.07e4i)T + (1.10e10 + 1.91e10i)T^{2} \)
59 \( 1 + (1.27e5 + 7.36e4i)T + (2.10e10 + 3.65e10i)T^{2} \)
61 \( 1 + (-6.34e3 - 1.09e4i)T + (-2.57e10 + 4.46e10i)T^{2} \)
67 \( 1 + (1.88e5 - 3.26e5i)T + (-4.52e10 - 7.83e10i)T^{2} \)
71 \( 1 + 1.18e5iT - 1.28e11T^{2} \)
73 \( 1 + (-3.41e3 + 5.91e3i)T + (-7.56e10 - 1.31e11i)T^{2} \)
79 \( 1 + (1.83e5 + 3.18e5i)T + (-1.21e11 + 2.10e11i)T^{2} \)
83 \( 1 + 3.36e5iT - 3.26e11T^{2} \)
89 \( 1 + (2.58e5 - 1.49e5i)T + (2.48e11 - 4.30e11i)T^{2} \)
97 \( 1 + 5.39e5T + 8.32e11T^{2} \)
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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

−17.30641158794401892795989632530, −15.53621175650252969721151866625, −14.24875611886373423721609206265, −13.10575211663651481315130142464, −12.01354386525823759371149796483, −9.253960138032233117195606107673, −8.942885450806257581088983923313, −7.10429105552012503605099913537, −4.48030527365482307768738654743, −2.22462525236433946725546285920, 1.64572174899549192799018323268, 4.18096523334168744654207269091, 6.47766357304740580504230923033, 8.569088263733361217151263502403, 9.793226120587080500461436294553, 10.87004799822555233065875296731, 13.38960286810668241890333401400, 14.23124812665722311021018100624, 14.89022266419430283401667087914, 16.90436967899541024421370995086

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