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Properties

Label 2-21-21.20-c21-0-19
Degree $2$
Conductor $21$
Sign $-0.817 + 0.575i$
Analytic cond. $58.6902$
Root an. cond. $7.66095$
Motivic weight $21$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.82e3i·2-s + (−2.08e4 − 1.00e5i)3-s − 5.86e6·4-s + 4.54e6·5-s + (−2.82e8 + 5.88e7i)6-s + (2.96e8 + 6.86e8i)7-s + 1.06e10i·8-s + (−9.59e9 + 4.17e9i)9-s − 1.28e10i·10-s + 7.30e10i·11-s + (1.22e11 + 5.87e11i)12-s − 3.14e11i·13-s + (1.93e12 − 8.36e11i)14-s + (−9.48e10 − 4.55e11i)15-s + 1.77e13·16-s − 1.38e13·17-s + ⋯
L(s)  = 1  − 1.94i·2-s + (−0.203 − 0.979i)3-s − 2.79·4-s + 0.208·5-s + (−1.90 + 0.397i)6-s + (0.396 + 0.917i)7-s + 3.50i·8-s + (−0.916 + 0.399i)9-s − 0.405i·10-s + 0.849i·11-s + (0.570 + 2.74i)12-s − 0.632i·13-s + (1.78 − 0.773i)14-s + (−0.0424 − 0.203i)15-s + 4.03·16-s − 1.66·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.817 + 0.575i)\, \overline{\Lambda}(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & (-0.817 + 0.575i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $-0.817 + 0.575i$
Analytic conductor: \(58.6902\)
Root analytic conductor: \(7.66095\)
Motivic weight: \(21\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :21/2),\ -0.817 + 0.575i)\)

Particular Values

\(L(11)\) \(\approx\) \(1.176859683\)
\(L(\frac12)\) \(\approx\) \(1.176859683\)
\(L(\frac{23}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.08e4 + 1.00e5i)T \)
7 \( 1 + (-2.96e8 - 6.86e8i)T \)
good2 \( 1 + 2.82e3iT - 2.09e6T^{2} \)
5 \( 1 - 4.54e6T + 4.76e14T^{2} \)
11 \( 1 - 7.30e10iT - 7.40e21T^{2} \)
13 \( 1 + 3.14e11iT - 2.47e23T^{2} \)
17 \( 1 + 1.38e13T + 6.90e25T^{2} \)
19 \( 1 - 1.99e13iT - 7.14e26T^{2} \)
23 \( 1 + 2.36e14iT - 3.94e28T^{2} \)
29 \( 1 + 4.13e14iT - 5.13e30T^{2} \)
31 \( 1 + 4.96e15iT - 2.08e31T^{2} \)
37 \( 1 - 1.68e16T + 8.55e32T^{2} \)
41 \( 1 - 3.82e16T + 7.38e33T^{2} \)
43 \( 1 - 1.23e17T + 2.00e34T^{2} \)
47 \( 1 - 5.21e17T + 1.30e35T^{2} \)
53 \( 1 + 4.78e17iT - 1.62e36T^{2} \)
59 \( 1 - 4.67e18T + 1.54e37T^{2} \)
61 \( 1 - 6.18e18iT - 3.10e37T^{2} \)
67 \( 1 + 4.47e18T + 2.22e38T^{2} \)
71 \( 1 + 1.47e19iT - 7.52e38T^{2} \)
73 \( 1 + 3.45e18iT - 1.34e39T^{2} \)
79 \( 1 - 9.44e19T + 7.08e39T^{2} \)
83 \( 1 - 4.40e19T + 1.99e40T^{2} \)
89 \( 1 + 1.20e20T + 8.65e40T^{2} \)
97 \( 1 - 8.64e20iT - 5.27e41T^{2} \)
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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

−12.57625920091638838613013549483, −11.76318014571079843506509262868, −10.68193142135865100936524359183, −9.225756545722853910993615281272, −8.094160331962721979418927049642, −5.75812594530930945415831962164, −4.36597666682576328031906403698, −2.42209447446922823094534363153, −2.07932335621533198765888765083, −0.65204475221438999322395954172, 0.53339517054168205548982413296, 3.85880349162413157322968706054, 4.73404826510868352097193543736, 5.93429822329113368003035127809, 7.11796814676488717411167354099, 8.577990547254332919222147174046, 9.486015048535852700368879378664, 11.02376923735992771999409905875, 13.51510999706496937766682207762, 14.14153647772913461967974709177

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