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Properties

Label 2-21-21.11-c32-0-30
Degree $2$
Conductor $21$
Sign $-0.706 - 0.707i$
Analytic cond. $136.219$
Root an. cond. $11.6713$
Motivic weight $32$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.15e7 + 3.72e7i)3-s + (−2.14e9 + 3.71e9i)4-s + (3.25e13 + 6.57e12i)7-s + (−9.26e14 − 1.60e15i)9-s + (−9.24e16 − 1.60e17i)12-s + 1.22e17·13-s + (−9.22e18 − 1.59e19i)16-s + (3.91e18 + 6.77e18i)19-s + (−9.46e20 + 1.07e21i)21-s + (−1.16e22 + 2.01e22i)25-s + 7.97e22·27-s + (−9.44e22 + 1.07e23i)28-s + (7.06e23 − 1.22e24i)31-s + (7.95e24 + 5.36e8i)36-s + (1.07e25 + 1.86e25i)37-s + ⋯
L(s)  = 1  + (−0.499 + 0.866i)3-s + (−0.5 + 0.866i)4-s + (0.980 + 0.197i)7-s + (−0.5 − 0.866i)9-s + (−0.499 − 0.866i)12-s + 0.183·13-s + (−0.499 − 0.866i)16-s + (0.0135 + 0.0234i)19-s + (−0.661 + 0.749i)21-s + (−0.5 + 0.866i)25-s + 27-s + (−0.661 + 0.749i)28-s + (0.971 − 1.68i)31-s + 36-s + (0.873 + 1.51i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.706 - 0.707i)\, \overline{\Lambda}(33-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+16) \, L(s)\cr =\mathstrut & (-0.706 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $-0.706 - 0.707i$
Analytic conductor: \(136.219\)
Root analytic conductor: \(11.6713\)
Motivic weight: \(32\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :16),\ -0.706 - 0.707i)\)

Particular Values

\(L(\frac{33}{2})\) \(\approx\) \(1.621370663\)
\(L(\frac12)\) \(\approx\) \(1.621370663\)
\(L(17)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.15e7 - 3.72e7i)T \)
7 \( 1 + (-3.25e13 - 6.57e12i)T \)
good2 \( 1 + (2.14e9 - 3.71e9i)T^{2} \)
5 \( 1 + (1.16e22 - 2.01e22i)T^{2} \)
11 \( 1 + (1.05e33 + 1.82e33i)T^{2} \)
13 \( 1 - 1.22e17T + 4.42e35T^{2} \)
17 \( 1 + (1.18e39 + 2.05e39i)T^{2} \)
19 \( 1 + (-3.91e18 - 6.77e18i)T + (-4.15e40 + 7.20e40i)T^{2} \)
23 \( 1 + (1.88e43 - 3.25e43i)T^{2} \)
29 \( 1 - 6.26e46T^{2} \)
31 \( 1 + (-7.06e23 + 1.22e24i)T + (-2.64e47 - 4.58e47i)T^{2} \)
37 \( 1 + (-1.07e25 - 1.86e25i)T + (-7.61e49 + 1.31e50i)T^{2} \)
41 \( 1 - 4.06e51T^{2} \)
43 \( 1 - 2.72e26T + 1.86e52T^{2} \)
47 \( 1 + (1.60e53 - 2.78e53i)T^{2} \)
53 \( 1 + (7.51e54 + 1.30e55i)T^{2} \)
59 \( 1 + (2.32e56 + 4.02e56i)T^{2} \)
61 \( 1 + (-1.97e28 - 3.42e28i)T + (-6.75e56 + 1.16e57i)T^{2} \)
67 \( 1 + (2.65e27 - 4.60e27i)T + (-1.35e58 - 2.35e58i)T^{2} \)
71 \( 1 - 1.73e59T^{2} \)
73 \( 1 + (3.83e29 - 6.64e29i)T + (-2.11e59 - 3.66e59i)T^{2} \)
79 \( 1 + (1.01e30 + 1.75e30i)T + (-2.64e60 + 4.58e60i)T^{2} \)
83 \( 1 - 2.57e61T^{2} \)
89 \( 1 + (1.20e62 - 2.07e62i)T^{2} \)
97 \( 1 - 1.21e32T + 3.77e63T^{2} \)
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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

−11.93123616594753840638631511863, −11.22404101111873193513511153439, −9.761494789861261556376771884386, −8.689233516055067288426319978044, −7.63279756437076862540639556802, −5.88955765549239595922325814973, −4.70701965937614222843312961314, −3.95482219675028789688841558724, −2.63899092748524125086599709242, −0.879249101903780611763628975886, 0.48501811786860854016447992974, 1.25463829816007482329923375189, 2.27803599298492874428520824623, 4.34519796733871313121492236275, 5.34329495517252849127077522964, 6.33967424979526184532554768217, 7.67351451001015672166724938868, 8.795514513826156545547640106280, 10.38839585772668743983918857480, 11.25977208872065284165053789426

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