Properties

Label 2-21-21.20-c29-0-4
Degree $2$
Conductor $21$
Sign $0.793 - 0.609i$
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  − 4.26e4i·2-s + (−4.19e6 + 7.14e6i)3-s − 1.28e9·4-s − 1.62e10·5-s + (3.04e11 + 1.78e11i)6-s + (1.66e12 + 6.74e11i)7-s + 3.16e13i·8-s + (−3.35e13 − 5.98e13i)9-s + 6.92e14i·10-s + 1.89e15i·11-s + (5.36e15 − 9.14e15i)12-s − 1.67e16i·13-s + (2.87e16 − 7.08e16i)14-s + (6.80e16 − 1.16e17i)15-s + 6.63e17·16-s + 1.70e17·17-s + ⋯
L(s)  = 1  − 1.83i·2-s + (−0.505 + 0.862i)3-s − 2.38·4-s − 1.18·5-s + (1.58 + 0.930i)6-s + (0.926 + 0.376i)7-s + 2.54i·8-s + (−0.488 − 0.872i)9-s + 2.18i·10-s + 1.50i·11-s + (1.20 − 2.05i)12-s − 1.17i·13-s + (0.691 − 1.70i)14-s + (0.601 − 1.02i)15-s + 2.30·16-s + 0.246·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.793 - 0.609i)\, \overline{\Lambda}(30-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+29/2) \, L(s)\cr =\mathstrut & (0.793 - 0.609i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $0.793 - 0.609i$
Analytic conductor: \(111.883\)
Root analytic conductor: \(10.5775\)
Motivic weight: \(29\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :29/2),\ 0.793 - 0.609i)\)

Particular Values

\(L(15)\) \(\approx\) \(0.3558104268\)
\(L(\frac12)\) \(\approx\) \(0.3558104268\)
\(L(\frac{31}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (4.19e6 - 7.14e6i)T \)
7 \( 1 + (-1.66e12 - 6.74e11i)T \)
good2 \( 1 + 4.26e4iT - 5.36e8T^{2} \)
5 \( 1 + 1.62e10T + 1.86e20T^{2} \)
11 \( 1 - 1.89e15iT - 1.58e30T^{2} \)
13 \( 1 + 1.67e16iT - 2.01e32T^{2} \)
17 \( 1 - 1.70e17T + 4.81e35T^{2} \)
19 \( 1 + 2.87e18iT - 1.21e37T^{2} \)
23 \( 1 + 8.58e18iT - 3.09e39T^{2} \)
29 \( 1 - 1.33e21iT - 2.56e42T^{2} \)
31 \( 1 + 8.21e21iT - 1.77e43T^{2} \)
37 \( 1 + 6.15e22T + 3.00e45T^{2} \)
41 \( 1 + 2.49e21T + 5.89e46T^{2} \)
43 \( 1 - 8.52e23T + 2.34e47T^{2} \)
47 \( 1 + 2.18e24T + 3.09e48T^{2} \)
53 \( 1 + 1.34e25iT - 1.00e50T^{2} \)
59 \( 1 + 2.12e25T + 2.26e51T^{2} \)
61 \( 1 - 5.23e25iT - 5.95e51T^{2} \)
67 \( 1 - 1.73e26T + 9.04e52T^{2} \)
71 \( 1 + 1.36e26iT - 4.85e53T^{2} \)
73 \( 1 + 1.03e27iT - 1.08e54T^{2} \)
79 \( 1 - 1.92e27T + 1.07e55T^{2} \)
83 \( 1 - 5.55e27T + 4.50e55T^{2} \)
89 \( 1 + 3.90e26T + 3.40e56T^{2} \)
97 \( 1 + 2.98e28iT - 4.13e57T^{2} \)
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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

−11.91464948083378224798591144835, −11.17667116554409534216080362809, −10.26248358566732063042771646034, −9.120309121201928702350709442894, −7.82461125145439751098607272003, −5.14350559489191649534350833482, −4.44718410891944980724177410635, −3.49511536470374536190587886667, −2.25290897487018907670316255037, −0.78875861067431748034924401790, 0.12886907823994791735238570373, 1.21132820499988768249398053612, 3.79823462027967811025976150551, 4.94897691326107572986839121494, 6.07415045368522080114097118621, 7.15153561649810319256643117079, 7.977391668831662361096755642011, 8.642033501801285851347442393751, 11.03890127293842724540596736398, 12.15370990770227465787175172789

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