Properties

Label 2-21-21.5-c29-0-10
Degree $2$
Conductor $21$
Sign $0.800 - 0.599i$
Analytic cond. $111.883$
Root an. cond. $10.5775$
Motivic weight $29$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (7.17e6 − 4.14e6i)3-s + (−2.68e8 − 4.64e8i)4-s + (−1.24e12 − 1.28e12i)7-s + (3.43e13 − 5.94e13i)9-s + (−3.85e15 − 2.22e15i)12-s + 1.93e16i·13-s + (−1.44e17 + 2.49e17i)16-s + (−1.96e18 − 1.13e18i)19-s + (−1.42e19 − 4.07e18i)21-s + (9.31e19 + 1.61e20i)25-s − 5.68e20i·27-s + (−2.64e20 + 9.26e20i)28-s + (−4.74e21 + 2.73e21i)31-s − 3.68e22·36-s + (−3.75e22 + 6.50e22i)37-s + ⋯
L(s)  = 1  + (0.866 − 0.499i)3-s + (−0.5 − 0.866i)4-s + (−0.695 − 0.718i)7-s + (0.5 − 0.866i)9-s + (−0.866 − 0.499i)12-s + 1.36i·13-s + (−0.499 + 0.866i)16-s + (−0.563 − 0.325i)19-s + (−0.961 − 0.274i)21-s + (0.5 + 0.866i)25-s − 0.999i·27-s + (−0.274 + 0.961i)28-s + (−1.12 + 0.649i)31-s − 0.999·36-s + (−0.685 + 1.18i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.800 - 0.599i)\, \overline{\Lambda}(30-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+29/2) \, L(s)\cr =\mathstrut & (0.800 - 0.599i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $0.800 - 0.599i$
Analytic conductor: \(111.883\)
Root analytic conductor: \(10.5775\)
Motivic weight: \(29\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :29/2),\ 0.800 - 0.599i)\)

Particular Values

\(L(15)\) \(\approx\) \(1.248064666\)
\(L(\frac12)\) \(\approx\) \(1.248064666\)
\(L(\frac{31}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-7.17e6 + 4.14e6i)T \)
7 \( 1 + (1.24e12 + 1.28e12i)T \)
good2 \( 1 + (2.68e8 + 4.64e8i)T^{2} \)
5 \( 1 + (-9.31e19 - 1.61e20i)T^{2} \)
11 \( 1 + (7.93e29 - 1.37e30i)T^{2} \)
13 \( 1 - 1.93e16iT - 2.01e32T^{2} \)
17 \( 1 + (-2.40e35 + 4.17e35i)T^{2} \)
19 \( 1 + (1.96e18 + 1.13e18i)T + (6.06e36 + 1.05e37i)T^{2} \)
23 \( 1 + (1.54e39 + 2.67e39i)T^{2} \)
29 \( 1 - 2.56e42T^{2} \)
31 \( 1 + (4.74e21 - 2.73e21i)T + (8.88e42 - 1.53e43i)T^{2} \)
37 \( 1 + (3.75e22 - 6.50e22i)T + (-1.50e45 - 2.60e45i)T^{2} \)
41 \( 1 + 5.89e46T^{2} \)
43 \( 1 - 9.11e23T + 2.34e47T^{2} \)
47 \( 1 + (-1.54e48 - 2.68e48i)T^{2} \)
53 \( 1 + (5.04e49 - 8.74e49i)T^{2} \)
59 \( 1 + (-1.13e51 + 1.95e51i)T^{2} \)
61 \( 1 + (9.20e25 + 5.31e25i)T + (2.97e51 + 5.15e51i)T^{2} \)
67 \( 1 + (1.75e26 + 3.04e26i)T + (-4.52e52 + 7.82e52i)T^{2} \)
71 \( 1 - 4.85e53T^{2} \)
73 \( 1 + (-1.71e27 + 9.87e26i)T + (5.43e53 - 9.41e53i)T^{2} \)
79 \( 1 + (3.19e27 - 5.53e27i)T + (-5.37e54 - 9.30e54i)T^{2} \)
83 \( 1 + 4.50e55T^{2} \)
89 \( 1 + (-1.70e56 - 2.95e56i)T^{2} \)
97 \( 1 + 7.63e28iT - 4.13e57T^{2} \)
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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

−12.63222986843902457616171516759, −10.85624882493359675891705533243, −9.558920341612446134062107517082, −8.865638993445539886912540829077, −7.20341929594517007233573798725, −6.32325682250160204156855606704, −4.57670981442294377327288041242, −3.48445742731489478000620941931, −1.94931114295699152049476924170, −0.979921407308752281584863247925, 0.26215381223237131854286688775, 2.35801538348285673396431960736, 3.18983948092927640249387278785, 4.18581397248839298245589176358, 5.61323995524880766182069813097, 7.45775076945606001523149816283, 8.492995520947261655588718357324, 9.318498562466087273289219318692, 10.55676556927689684389961847797, 12.44338508248290124982005653343

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