Properties

Label 2-21-7.2-c29-0-9
Degree $2$
Conductor $21$
Sign $-0.139 + 0.990i$
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  + (−5.31e3 + 9.20e3i)2-s + (2.39e6 + 4.14e6i)3-s + (2.11e8 + 3.67e8i)4-s + (6.21e8 − 1.07e9i)5-s + (−5.08e10 + 3.81e−6i)6-s + (−1.19e12 + 1.34e12i)7-s + (−1.02e13 + 0.000732i)8-s + (−1.14e13 + 1.98e13i)9-s + (6.59e12 + 1.14e13i)10-s + (6.51e14 + 1.12e15i)11-s + (−1.01e15 + 1.75e15i)12-s − 2.09e16·13-s + (−5.99e15 − 1.80e16i)14-s + 5.94e15·15-s + (−5.95e16 + 1.03e17i)16-s + (6.30e17 + 1.09e18i)17-s + ⋯
L(s)  = 1  + (−0.229 + 0.397i)2-s + (0.288 + 0.499i)3-s + (0.394 + 0.683i)4-s + (0.0455 − 0.0788i)5-s − 0.264·6-s + (−0.664 + 0.746i)7-s − 0.820·8-s + (−0.166 + 0.288i)9-s + (0.0208 + 0.0361i)10-s + (0.516 + 0.895i)11-s + (−0.227 + 0.394i)12-s − 1.47·13-s + (−0.144 − 0.435i)14-s + 0.0525·15-s + (−0.206 + 0.357i)16-s + (0.908 + 1.57i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.139 + 0.990i)\, \overline{\Lambda}(30-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+29/2) \, L(s)\cr =\mathstrut & (-0.139 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $-0.139 + 0.990i$
Analytic conductor: \(111.883\)
Root analytic conductor: \(10.5775\)
Motivic weight: \(29\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :29/2),\ -0.139 + 0.990i)\)

Particular Values

\(L(15)\) \(\approx\) \(1.357280826\)
\(L(\frac12)\) \(\approx\) \(1.357280826\)
\(L(\frac{31}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.39e6 - 4.14e6i)T \)
7 \( 1 + (1.19e12 - 1.34e12i)T \)
good2 \( 1 + (5.31e3 - 9.20e3i)T + (-2.68e8 - 4.64e8i)T^{2} \)
5 \( 1 + (-6.21e8 + 1.07e9i)T + (-9.31e19 - 1.61e20i)T^{2} \)
11 \( 1 + (-6.51e14 - 1.12e15i)T + (-7.93e29 + 1.37e30i)T^{2} \)
13 \( 1 + 2.09e16T + 2.01e32T^{2} \)
17 \( 1 + (-6.30e17 - 1.09e18i)T + (-2.40e35 + 4.17e35i)T^{2} \)
19 \( 1 + (2.32e18 - 4.03e18i)T + (-6.06e36 - 1.05e37i)T^{2} \)
23 \( 1 + (-2.15e19 + 3.72e19i)T + (-1.54e39 - 2.67e39i)T^{2} \)
29 \( 1 - 3.48e20T + 2.56e42T^{2} \)
31 \( 1 + (6.24e20 + 1.08e21i)T + (-8.88e42 + 1.53e43i)T^{2} \)
37 \( 1 + (1.27e22 - 2.19e22i)T + (-1.50e45 - 2.60e45i)T^{2} \)
41 \( 1 + 8.92e22T + 5.89e46T^{2} \)
43 \( 1 - 2.21e23T + 2.34e47T^{2} \)
47 \( 1 + (-3.16e23 + 5.48e23i)T + (-1.54e48 - 2.68e48i)T^{2} \)
53 \( 1 + (-9.53e24 - 1.65e25i)T + (-5.04e49 + 8.74e49i)T^{2} \)
59 \( 1 + (9.83e24 + 1.70e25i)T + (-1.13e51 + 1.95e51i)T^{2} \)
61 \( 1 + (5.52e25 - 9.57e25i)T + (-2.97e51 - 5.15e51i)T^{2} \)
67 \( 1 + (3.68e25 + 6.37e25i)T + (-4.52e52 + 7.82e52i)T^{2} \)
71 \( 1 - 1.06e27T + 4.85e53T^{2} \)
73 \( 1 + (-1.92e26 - 3.33e26i)T + (-5.43e53 + 9.41e53i)T^{2} \)
79 \( 1 + (-1.49e26 + 2.58e26i)T + (-5.37e54 - 9.30e54i)T^{2} \)
83 \( 1 + 4.63e27T + 4.50e55T^{2} \)
89 \( 1 + (5.18e27 - 8.97e27i)T + (-1.70e56 - 2.95e56i)T^{2} \)
97 \( 1 - 3.61e28T + 4.13e57T^{2} \)
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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

−12.57943217049510003467222337649, −12.21752825466069721373485548699, −10.31316699697480152420087865297, −9.255897748669951540368048444791, −8.166591959727117173704491297695, −6.96301804916866808141835385827, −5.73323058073252461442509124011, −4.13658404042747909124094490190, −2.98155121867514013274515132338, −1.90775732590019086653574049819, 0.35032379455499394719231626341, 0.852588627464548212985853843031, 2.38396755793452884362253657694, 3.21389656357134419329159220250, 5.07465193475485189489886118093, 6.54258653785321588867763526164, 7.30758070710502454068930637638, 9.115792854054421369420796203703, 9.993036764412519584193017687366, 11.24128183432758637951173941314

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