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Properties

Label 2-21-3.2-c6-0-9
Degree $2$
Conductor $21$
Sign $-0.371 + 0.928i$
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  − 5.05i·2-s + (10.0 − 25.0i)3-s + 38.4·4-s − 24.8i·5-s + (−126. − 50.7i)6-s − 129.·7-s − 518. i·8-s + (−527. − 503. i)9-s − 125.·10-s + 230. i·11-s + (385. − 962. i)12-s + 690.·13-s + 655. i·14-s + (−622. − 249. i)15-s − 161.·16-s + 4.80e3i·17-s + ⋯
L(s)  = 1  − 0.632i·2-s + (0.371 − 0.928i)3-s + 0.600·4-s − 0.198i·5-s + (−0.586 − 0.235i)6-s − 0.377·7-s − 1.01i·8-s + (−0.723 − 0.690i)9-s − 0.125·10-s + 0.173i·11-s + (0.223 − 0.557i)12-s + 0.314·13-s + 0.238i·14-s + (−0.184 − 0.0738i)15-s − 0.0394·16-s + 0.978i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.371 + 0.928i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.371 + 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $-0.371 + 0.928i$
Analytic conductor: \(4.83113\)
Root analytic conductor: \(2.19798\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :3),\ -0.371 + 0.928i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.04769 - 1.54815i\)
\(L(\frac12)\) \(\approx\) \(1.04769 - 1.54815i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-10.0 + 25.0i)T \)
7 \( 1 + 129.T \)
good2 \( 1 + 5.05iT - 64T^{2} \)
5 \( 1 + 24.8iT - 1.56e4T^{2} \)
11 \( 1 - 230. iT - 1.77e6T^{2} \)
13 \( 1 - 690.T + 4.82e6T^{2} \)
17 \( 1 - 4.80e3iT - 2.41e7T^{2} \)
19 \( 1 - 1.11e4T + 4.70e7T^{2} \)
23 \( 1 - 8.67e3iT - 1.48e8T^{2} \)
29 \( 1 + 2.42e4iT - 5.94e8T^{2} \)
31 \( 1 + 7.15e3T + 8.87e8T^{2} \)
37 \( 1 - 2.95e4T + 2.56e9T^{2} \)
41 \( 1 - 1.02e5iT - 4.75e9T^{2} \)
43 \( 1 + 1.08e5T + 6.32e9T^{2} \)
47 \( 1 - 2.03e5iT - 1.07e10T^{2} \)
53 \( 1 + 3.51e4iT - 2.21e10T^{2} \)
59 \( 1 + 2.24e5iT - 4.21e10T^{2} \)
61 \( 1 + 3.54e4T + 5.15e10T^{2} \)
67 \( 1 + 5.23e5T + 9.04e10T^{2} \)
71 \( 1 + 6.22e5iT - 1.28e11T^{2} \)
73 \( 1 + 9.99e4T + 1.51e11T^{2} \)
79 \( 1 - 2.75e5T + 2.43e11T^{2} \)
83 \( 1 - 3.75e5iT - 3.26e11T^{2} \)
89 \( 1 - 6.17e5iT - 4.96e11T^{2} \)
97 \( 1 + 1.14e6T + 8.32e11T^{2} \)
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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

−16.47497571437292266457385953962, −15.08343999216347933981497610598, −13.45495433633204693359441904609, −12.44161819732458404333768873851, −11.32559373920816084758486735187, −9.591387863896811935883244607436, −7.74646694162602800941433637505, −6.29145370741519922490429987360, −3.15508755600966401002801477697, −1.33912215542198745202664543877, 3.02622517914671722806383721871, 5.35118093214802912043057852106, 7.15049047303824441802876202310, 8.815181649672686515552723819096, 10.38658062324198360837357821661, 11.67378652421238297433237551784, 13.83601156947243866516557204319, 14.91386724504604122274896884340, 16.04838504036724622163425771682, 16.60165136167440597959739982826

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