Properties

Label 2-21-7.5-c6-0-0
Degree $2$
Conductor $21$
Sign $-0.796 + 0.605i$
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  + (−2.76 + 4.78i)2-s + (−13.5 + 7.79i)3-s + (16.7 + 28.9i)4-s + (−57.9 − 33.4i)5-s − 86.1i·6-s + (−240. − 244. i)7-s − 538.·8-s + (121.5 − 210. i)9-s + (320. − 185. i)10-s + (−862. − 1.49e3i)11-s + (−451. − 260. i)12-s + 2.80e3i·13-s + (1.83e3 − 475. i)14-s + 1.04e3·15-s + (418. − 724. i)16-s + (−5.32e3 + 3.07e3i)17-s + ⋯
L(s)  = 1  + (−0.345 + 0.598i)2-s + (−0.5 + 0.288i)3-s + (0.261 + 0.452i)4-s + (−0.463 − 0.267i)5-s − 0.398i·6-s + (−0.701 − 0.713i)7-s − 1.05·8-s + (0.166 − 0.288i)9-s + (0.320 − 0.185i)10-s + (−0.648 − 1.12i)11-s + (−0.261 − 0.150i)12-s + 1.27i·13-s + (0.668 − 0.173i)14-s + 0.309·15-s + (0.102 − 0.176i)16-s + (−1.08 + 0.625i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.796 + 0.605i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.796 + 0.605i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $-0.796 + 0.605i$
Analytic conductor: \(4.83113\)
Root analytic conductor: \(2.19798\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :3),\ -0.796 + 0.605i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.0596260 - 0.176928i\)
\(L(\frac12)\) \(\approx\) \(0.0596260 - 0.176928i\)
\(L(4)\) 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 \( 1 + (240. + 244. i)T \)
good2 \( 1 + (2.76 - 4.78i)T + (-32 - 55.4i)T^{2} \)
5 \( 1 + (57.9 + 33.4i)T + (7.81e3 + 1.35e4i)T^{2} \)
11 \( 1 + (862. + 1.49e3i)T + (-8.85e5 + 1.53e6i)T^{2} \)
13 \( 1 - 2.80e3iT - 4.82e6T^{2} \)
17 \( 1 + (5.32e3 - 3.07e3i)T + (1.20e7 - 2.09e7i)T^{2} \)
19 \( 1 + (-7.73e3 - 4.46e3i)T + (2.35e7 + 4.07e7i)T^{2} \)
23 \( 1 + (4.95e3 - 8.57e3i)T + (-7.40e7 - 1.28e8i)T^{2} \)
29 \( 1 + 1.36e4T + 5.94e8T^{2} \)
31 \( 1 + (2.18e4 - 1.26e4i)T + (4.43e8 - 7.68e8i)T^{2} \)
37 \( 1 + (1.13e4 - 1.96e4i)T + (-1.28e9 - 2.22e9i)T^{2} \)
41 \( 1 + 3.78e4iT - 4.75e9T^{2} \)
43 \( 1 - 7.36e4T + 6.32e9T^{2} \)
47 \( 1 + (-1.20e5 - 6.95e4i)T + (5.38e9 + 9.33e9i)T^{2} \)
53 \( 1 + (-8.25e3 - 1.43e4i)T + (-1.10e10 + 1.91e10i)T^{2} \)
59 \( 1 + (6.19e4 - 3.57e4i)T + (2.10e10 - 3.65e10i)T^{2} \)
61 \( 1 + (2.28e5 + 1.31e5i)T + (2.57e10 + 4.46e10i)T^{2} \)
67 \( 1 + (2.74e5 + 4.75e5i)T + (-4.52e10 + 7.83e10i)T^{2} \)
71 \( 1 - 4.65e5T + 1.28e11T^{2} \)
73 \( 1 + (2.89e5 - 1.66e5i)T + (7.56e10 - 1.31e11i)T^{2} \)
79 \( 1 + (-2.05e5 + 3.55e5i)T + (-1.21e11 - 2.10e11i)T^{2} \)
83 \( 1 - 7.19e4iT - 3.26e11T^{2} \)
89 \( 1 + (1.41e5 + 8.16e4i)T + (2.48e11 + 4.30e11i)T^{2} \)
97 \( 1 - 6.51e5iT - 8.32e11T^{2} \)
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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

−17.19167346997020197826579234769, −16.23719957122484010686127046635, −15.78082405908236448334012650172, −13.74505750249019307226426246585, −12.20637771806521707076266332608, −10.96272149688307756491253366552, −9.142094730630209064625482761074, −7.58838686491116326353844473490, −6.15600318161202427310998481392, −3.76214936758483888524351914236, 0.13003306221920434091949669750, 2.57102049400270543897504552507, 5.54566399907551456002992985096, 7.23122827869277078580575861095, 9.419935737028294501204421675218, 10.68547472855786870154883502280, 11.88441672589192859068209451728, 12.99103754814306468963072033154, 15.19573955171648518520809656251, 15.78816168372341428031146160422

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