Properties

Label 2-21-21.2-c28-0-56
Degree $2$
Conductor $21$
Sign $-0.931 - 0.364i$
Analytic cond. $104.303$
Root an. cond. $10.2129$
Motivic weight $28$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.39e6 − 4.14e6i)3-s + (−1.34e8 − 2.32e8i)4-s + (3.89e11 + 5.55e11i)7-s + (−1.14e13 + 1.98e13i)9-s + (−6.41e14 + 1.11e15i)12-s + 6.87e15·13-s + (−3.60e16 + 6.24e16i)16-s + (−4.51e17 + 7.82e17i)19-s + (1.37e18 − 2.94e18i)21-s + (−1.86e19 − 3.22e19i)25-s + (1.09e20 − 8.19e3i)27-s + (7.68e19 − 1.65e20i)28-s + (−7.44e20 − 1.28e21i)31-s + (6.14e21 − 5.24e5i)36-s + (6.03e21 − 1.04e22i)37-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)3-s + (−0.5 − 0.866i)4-s + (0.573 + 0.819i)7-s + (−0.499 + 0.866i)9-s + (−0.499 + 0.866i)12-s + 1.74·13-s + (−0.499 + 0.866i)16-s + (−0.565 + 0.979i)19-s + (0.422 − 0.906i)21-s + (−0.5 − 0.866i)25-s + 27-s + (0.422 − 0.906i)28-s + (−0.983 − 1.70i)31-s + 0.999·36-s + (0.669 − 1.15i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.931 - 0.364i)\, \overline{\Lambda}(29-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+14) \, L(s)\cr =\mathstrut & (-0.931 - 0.364i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $-0.931 - 0.364i$
Analytic conductor: \(104.303\)
Root analytic conductor: \(10.2129\)
Motivic weight: \(28\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :14),\ -0.931 - 0.364i)\)

Particular Values

\(L(\frac{29}{2})\) \(\approx\) \(0.5072712744\)
\(L(\frac12)\) \(\approx\) \(0.5072712744\)
\(L(15)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.39e6 + 4.14e6i)T \)
7 \( 1 + (-3.89e11 - 5.55e11i)T \)
good2 \( 1 + (1.34e8 + 2.32e8i)T^{2} \)
5 \( 1 + (1.86e19 + 3.22e19i)T^{2} \)
11 \( 1 + (7.21e28 - 1.24e29i)T^{2} \)
13 \( 1 - 6.87e15T + 1.55e31T^{2} \)
17 \( 1 + (1.41e34 - 2.45e34i)T^{2} \)
19 \( 1 + (4.51e17 - 7.82e17i)T + (-3.19e35 - 5.52e35i)T^{2} \)
23 \( 1 + (6.71e37 + 1.16e38i)T^{2} \)
29 \( 1 - 8.85e40T^{2} \)
31 \( 1 + (7.44e20 + 1.28e21i)T + (-2.86e41 + 4.96e41i)T^{2} \)
37 \( 1 + (-6.03e21 + 1.04e22i)T + (-4.06e43 - 7.03e43i)T^{2} \)
41 \( 1 - 1.43e45T^{2} \)
43 \( 1 + 6.43e21T + 5.45e45T^{2} \)
47 \( 1 + (3.29e46 + 5.70e46i)T^{2} \)
53 \( 1 + (9.52e47 - 1.64e48i)T^{2} \)
59 \( 1 + (1.91e49 - 3.32e49i)T^{2} \)
61 \( 1 + (9.43e24 - 1.63e25i)T + (-4.87e49 - 8.44e49i)T^{2} \)
67 \( 1 + (-7.67e24 - 1.32e25i)T + (-6.74e50 + 1.16e51i)T^{2} \)
71 \( 1 - 6.84e51T^{2} \)
73 \( 1 + (8.94e25 + 1.54e26i)T + (-7.44e51 + 1.28e52i)T^{2} \)
79 \( 1 + (-2.06e26 + 3.58e26i)T + (-6.80e52 - 1.17e53i)T^{2} \)
83 \( 1 - 5.42e53T^{2} \)
89 \( 1 + (1.91e54 + 3.31e54i)T^{2} \)
97 \( 1 + 1.29e28T + 4.26e55T^{2} \)
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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

−11.54370486128746280167623697518, −10.61725803123886756213023121649, −8.960817714515674781069939160387, −7.961301172217900841832423389435, −6.08919488974275390680907888364, −5.73745651784384568198677077457, −4.19035164425144225049422445694, −2.12118093713911370843700142234, −1.29948465130286345742570753535, −0.13383214249565080815693335820, 1.17203519139096739324049184661, 3.29601253362586455047019551353, 4.08574377230321824985297617897, 5.11194860643855725247459759428, 6.70864255215723153107750929289, 8.229252514827666310030946348831, 9.197245163761349289297828314542, 10.72613567719885376904870517480, 11.47243957370491037938932419505, 13.00843820405041133151491727064

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