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Properties

Label 2-21-21.5-c7-0-15
Degree $2$
Conductor $21$
Sign $-0.358 + 0.933i$
Analytic cond. $6.56008$
Root an. cond. $2.56126$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (40.5 − 23.3i)3-s + (−64 − 110. i)4-s + (−881.5 − 215. i)7-s + (1.09e3 − 1.89e3i)9-s + (−5.18e3 − 2.99e3i)12-s − 1.57e4i·13-s + (−8.19e3 + 1.41e4i)16-s + (5.02e4 + 2.90e4i)19-s + (−4.07e4 + 1.18e4i)21-s + (3.90e4 + 6.76e4i)25-s − 1.02e5i·27-s + (3.25e4 + 1.11e5i)28-s + (−1.32e4 + 7.63e3i)31-s − 2.79e5·36-s + (1.67e5 − 2.90e5i)37-s + ⋯
L(s)  = 1  + (0.866 − 0.499i)3-s + (−0.5 − 0.866i)4-s + (−0.971 − 0.237i)7-s + (0.5 − 0.866i)9-s + (−0.866 − 0.499i)12-s − 1.98i·13-s + (−0.499 + 0.866i)16-s + (1.68 + 0.970i)19-s + (−0.960 + 0.279i)21-s + (0.5 + 0.866i)25-s − 0.999i·27-s + (0.279 + 0.960i)28-s + (−0.0797 + 0.0460i)31-s − 36-s + (0.544 − 0.943i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.358 + 0.933i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.358 + 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $-0.358 + 0.933i$
Analytic conductor: \(6.56008\)
Root analytic conductor: \(2.56126\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :7/2),\ -0.358 + 0.933i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.900244 - 1.31007i\)
\(L(\frac12)\) \(\approx\) \(0.900244 - 1.31007i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-40.5 + 23.3i)T \)
7 \( 1 + (881.5 + 215. i)T \)
good2 \( 1 + (64 + 110. i)T^{2} \)
5 \( 1 + (-3.90e4 - 6.76e4i)T^{2} \)
11 \( 1 + (9.74e6 - 1.68e7i)T^{2} \)
13 \( 1 + 1.57e4iT - 6.27e7T^{2} \)
17 \( 1 + (-2.05e8 + 3.55e8i)T^{2} \)
19 \( 1 + (-5.02e4 - 2.90e4i)T + (4.46e8 + 7.74e8i)T^{2} \)
23 \( 1 + (1.70e9 + 2.94e9i)T^{2} \)
29 \( 1 - 1.72e10T^{2} \)
31 \( 1 + (1.32e4 - 7.63e3i)T + (1.37e10 - 2.38e10i)T^{2} \)
37 \( 1 + (-1.67e5 + 2.90e5i)T + (-4.74e10 - 8.22e10i)T^{2} \)
41 \( 1 + 1.94e11T^{2} \)
43 \( 1 - 4.09e5T + 2.71e11T^{2} \)
47 \( 1 + (-2.53e11 - 4.38e11i)T^{2} \)
53 \( 1 + (5.87e11 - 1.01e12i)T^{2} \)
59 \( 1 + (-1.24e12 + 2.15e12i)T^{2} \)
61 \( 1 + (2.30e5 + 1.33e5i)T + (1.57e12 + 2.72e12i)T^{2} \)
67 \( 1 + (2.22e6 + 3.84e6i)T + (-3.03e12 + 5.24e12i)T^{2} \)
71 \( 1 - 9.09e12T^{2} \)
73 \( 1 + (5.65e6 - 3.26e6i)T + (5.52e12 - 9.56e12i)T^{2} \)
79 \( 1 + (-2.25e6 + 3.91e6i)T + (-9.60e12 - 1.66e13i)T^{2} \)
83 \( 1 + 2.71e13T^{2} \)
89 \( 1 + (-2.21e13 - 3.83e13i)T^{2} \)
97 \( 1 - 1.31e7iT - 8.07e13T^{2} \)
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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

−15.84587729529559055110518676044, −14.75088576243766584298235844614, −13.56468422893883351378953679894, −12.65286444095662741402176102182, −10.31771637685868452722618058280, −9.289107380849104356005087199310, −7.60970864584907729230283543163, −5.76586440869246166285909646254, −3.29542571803094258523090078040, −0.864707980054412244225787254744, 2.90567643428056068720684602728, 4.39426827804925410853316034942, 7.13851234781898857230060019209, 8.855315627395578771944507194949, 9.623078507633785008622775172761, 11.79874764689413849130128738174, 13.29016365468036665283568083573, 14.16148773160337675803143766646, 15.91019834058981815536799198426, 16.60287563790474235831036784836

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