Properties

Label 2-21-21.5-c5-0-4
Degree $2$
Conductor $21$
Sign $0.342 + 0.939i$
Analytic cond. $3.36806$
Root an. cond. $1.83522$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−8.95 − 5.16i)2-s + (14.3 + 6.20i)3-s + (37.4 + 64.7i)4-s + (15.8 − 27.4i)5-s + (−95.9 − 129. i)6-s + (20.8 − 127. i)7-s − 442. i·8-s + (166. + 177. i)9-s + (−284. + 164. i)10-s + (392. − 226. i)11-s + (133. + 1.15e3i)12-s − 551. i·13-s + (−847. + 1.03e3i)14-s + (397. − 294. i)15-s + (−1.08e3 + 1.88e3i)16-s + (269. + 466. i)17-s + ⋯
L(s)  = 1  + (−1.58 − 0.913i)2-s + (0.917 + 0.397i)3-s + (1.16 + 2.02i)4-s + (0.283 − 0.491i)5-s + (−1.08 − 1.46i)6-s + (0.160 − 0.986i)7-s − 2.44i·8-s + (0.683 + 0.729i)9-s + (−0.898 + 0.518i)10-s + (0.978 − 0.565i)11-s + (0.267 + 2.32i)12-s − 0.905i·13-s + (−1.15 + 1.41i)14-s + (0.456 − 0.338i)15-s + (−1.06 + 1.84i)16-s + (0.225 + 0.391i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 + 0.939i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $0.342 + 0.939i$
Analytic conductor: \(3.36806\)
Root analytic conductor: \(1.83522\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :5/2),\ 0.342 + 0.939i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.805584 - 0.563572i\)
\(L(\frac12)\) \(\approx\) \(0.805584 - 0.563572i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-14.3 - 6.20i)T \)
7 \( 1 + (-20.8 + 127. i)T \)
good2 \( 1 + (8.95 + 5.16i)T + (16 + 27.7i)T^{2} \)
5 \( 1 + (-15.8 + 27.4i)T + (-1.56e3 - 2.70e3i)T^{2} \)
11 \( 1 + (-392. + 226. i)T + (8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 + 551. iT - 3.71e5T^{2} \)
17 \( 1 + (-269. - 466. i)T + (-7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (-1.18e3 - 683. i)T + (1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (2.80e3 + 1.61e3i)T + (3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 - 1.60e3iT - 2.05e7T^{2} \)
31 \( 1 + (6.12e3 - 3.53e3i)T + (1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (-975. + 1.68e3i)T + (-3.46e7 - 6.00e7i)T^{2} \)
41 \( 1 + 1.80e3T + 1.15e8T^{2} \)
43 \( 1 - 7.88e3T + 1.47e8T^{2} \)
47 \( 1 + (3.04e3 - 5.27e3i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + (1.22e4 - 7.08e3i)T + (2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (-8.45e3 - 1.46e4i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (2.57e4 + 1.48e4i)T + (4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (6.80e3 + 1.17e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 - 3.13e4iT - 1.80e9T^{2} \)
73 \( 1 + (-8.50e3 + 4.91e3i)T + (1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (4.98e4 - 8.64e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 - 7.95e4T + 3.93e9T^{2} \)
89 \( 1 + (-476. + 825. i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 + 1.15e5iT - 8.58e9T^{2} \)
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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

−17.01344447179585801697532355505, −16.21444068108278788084583582848, −14.20317132634755294491385276841, −12.64911564841345485672760949953, −10.89099349748160781332201680230, −9.869655783799907875101785501800, −8.743500325897750475202343230112, −7.61044809746417111387992684339, −3.53300530021311626372580070326, −1.31269272249557027309153577429, 1.90843767445440765060805592616, 6.35072101469502090548940830580, 7.55503501354286476082334286278, 9.010632694251021822655842353755, 9.698150711385296703030472940726, 11.76784016588340107966673304901, 14.16625706300359290026736218958, 14.98956095350824903472222763229, 16.14661948557974217028676002750, 17.71977394918391372421197643258

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