Properties

Label 2-21-7.3-c6-0-6
Degree $2$
Conductor $21$
Sign $-0.505 + 0.862i$
Analytic cond. $4.83113$
Root an. cond. $2.19798$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.65 − 8.06i)2-s + (13.5 + 7.79i)3-s + (−11.3 + 19.6i)4-s + (151. − 87.3i)5-s − 145. i·6-s + (−271. − 209. i)7-s − 384.·8-s + (121.5 + 210. i)9-s + (−1.40e3 − 813. i)10-s + (92.6 − 160. i)11-s + (−305. + 176. i)12-s − 3.98e3i·13-s + (−424. + 3.16e3i)14-s + 2.72e3·15-s + (2.51e3 + 4.35e3i)16-s + (6.10e3 + 3.52e3i)17-s + ⋯
L(s)  = 1  + (−0.581 − 1.00i)2-s + (0.5 + 0.288i)3-s + (−0.176 + 0.306i)4-s + (1.21 − 0.699i)5-s − 0.671i·6-s + (−0.791 − 0.610i)7-s − 0.751·8-s + (0.166 + 0.288i)9-s + (−1.40 − 0.813i)10-s + (0.0696 − 0.120i)11-s + (−0.176 + 0.102i)12-s − 1.81i·13-s + (−0.154 + 1.15i)14-s + 0.807·15-s + (0.614 + 1.06i)16-s + (1.24 + 0.716i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.505 + 0.862i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.505 + 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $-0.505 + 0.862i$
Analytic conductor: \(4.83113\)
Root analytic conductor: \(2.19798\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :3),\ -0.505 + 0.862i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.706460 - 1.23296i\)
\(L(\frac12)\) \(\approx\) \(0.706460 - 1.23296i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-13.5 - 7.79i)T \)
7 \( 1 + (271. + 209. i)T \)
good2 \( 1 + (4.65 + 8.06i)T + (-32 + 55.4i)T^{2} \)
5 \( 1 + (-151. + 87.3i)T + (7.81e3 - 1.35e4i)T^{2} \)
11 \( 1 + (-92.6 + 160. i)T + (-8.85e5 - 1.53e6i)T^{2} \)
13 \( 1 + 3.98e3iT - 4.82e6T^{2} \)
17 \( 1 + (-6.10e3 - 3.52e3i)T + (1.20e7 + 2.09e7i)T^{2} \)
19 \( 1 + (4.06e3 - 2.34e3i)T + (2.35e7 - 4.07e7i)T^{2} \)
23 \( 1 + (-3.32e3 - 5.75e3i)T + (-7.40e7 + 1.28e8i)T^{2} \)
29 \( 1 - 1.93e4T + 5.94e8T^{2} \)
31 \( 1 + (-2.07e4 - 1.19e4i)T + (4.43e8 + 7.68e8i)T^{2} \)
37 \( 1 + (-1.97e4 - 3.42e4i)T + (-1.28e9 + 2.22e9i)T^{2} \)
41 \( 1 - 4.62e4iT - 4.75e9T^{2} \)
43 \( 1 + 9.31e4T + 6.32e9T^{2} \)
47 \( 1 + (-1.29e5 + 7.47e4i)T + (5.38e9 - 9.33e9i)T^{2} \)
53 \( 1 + (-5.09e4 + 8.81e4i)T + (-1.10e10 - 1.91e10i)T^{2} \)
59 \( 1 + (-1.31e5 - 7.61e4i)T + (2.10e10 + 3.65e10i)T^{2} \)
61 \( 1 + (1.45e5 - 8.42e4i)T + (2.57e10 - 4.46e10i)T^{2} \)
67 \( 1 + (3.05e4 - 5.29e4i)T + (-4.52e10 - 7.83e10i)T^{2} \)
71 \( 1 + 4.85e5T + 1.28e11T^{2} \)
73 \( 1 + (7.73e3 + 4.46e3i)T + (7.56e10 + 1.31e11i)T^{2} \)
79 \( 1 + (2.21e5 + 3.84e5i)T + (-1.21e11 + 2.10e11i)T^{2} \)
83 \( 1 - 5.59e5iT - 3.26e11T^{2} \)
89 \( 1 + (1.79e5 - 1.03e5i)T + (2.48e11 - 4.30e11i)T^{2} \)
97 \( 1 + 1.23e6iT - 8.32e11T^{2} \)
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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

−16.75009966213475529745844588161, −15.08875210130603258452530979576, −13.44657392240594873309610739126, −12.51436206579679159008124615531, −10.26922981570550744162039026848, −9.975446585185686289348671231333, −8.462672915140198393392114469279, −5.81124299874847215197604180979, −3.09802591702431119873927515125, −1.12389530671090047301397717855, 2.59187709771297644202042665241, 6.13910874393800482084465522634, 7.01677059026643105071608940861, 8.925530265620749234944812959570, 9.789403262925515602133770988915, 12.12781465633240036265484573496, 13.81250444304388767025469934863, 14.73608123889513936969192962618, 16.14296654351800889972093633895, 17.19607885889093517526385603164

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