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Properties

Label 2-21-21.20-c21-0-29
Degree $2$
Conductor $21$
Sign $0.697 + 0.716i$
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  + 1.92e3i·2-s + (−7.83e4 − 6.57e4i)3-s − 1.61e6·4-s − 2.77e7·5-s + (1.26e8 − 1.50e8i)6-s + (7.43e8 + 7.50e7i)7-s + 9.36e8i·8-s + (1.81e9 + 1.03e10i)9-s − 5.34e10i·10-s − 5.61e10i·11-s + (1.26e11 + 1.05e11i)12-s + 3.86e11i·13-s + (−1.44e11 + 1.43e12i)14-s + (2.17e12 + 1.82e12i)15-s − 5.18e12·16-s − 1.33e13·17-s + ⋯
L(s)  = 1  + 1.32i·2-s + (−0.766 − 0.642i)3-s − 0.768·4-s − 1.27·5-s + (0.854 − 1.01i)6-s + (0.994 + 0.100i)7-s + 0.308i·8-s + (0.173 + 0.984i)9-s − 1.69i·10-s − 0.652i·11-s + (0.588 + 0.493i)12-s + 0.777i·13-s + (−0.133 + 1.32i)14-s + (0.973 + 0.816i)15-s − 1.17·16-s − 1.60·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.697 + 0.716i)\, \overline{\Lambda}(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & (0.697 + 0.716i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $0.697 + 0.716i$
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: \((2,\ 21,\ (\ :21/2),\ 0.697 + 0.716i)\)

Particular Values

\(L(11)\) \(\approx\) \(0.1419185504\)
\(L(\frac12)\) \(\approx\) \(0.1419185504\)
\(L(\frac{23}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (7.83e4 + 6.57e4i)T \)
7 \( 1 + (-7.43e8 - 7.50e7i)T \)
good2 \( 1 - 1.92e3iT - 2.09e6T^{2} \)
5 \( 1 + 2.77e7T + 4.76e14T^{2} \)
11 \( 1 + 5.61e10iT - 7.40e21T^{2} \)
13 \( 1 - 3.86e11iT - 2.47e23T^{2} \)
17 \( 1 + 1.33e13T + 6.90e25T^{2} \)
19 \( 1 - 3.04e13iT - 7.14e26T^{2} \)
23 \( 1 - 2.80e14iT - 3.94e28T^{2} \)
29 \( 1 - 2.28e15iT - 5.13e30T^{2} \)
31 \( 1 + 3.90e15iT - 2.08e31T^{2} \)
37 \( 1 - 3.61e16T + 8.55e32T^{2} \)
41 \( 1 + 7.43e16T + 7.38e33T^{2} \)
43 \( 1 + 1.37e17T + 2.00e34T^{2} \)
47 \( 1 - 1.27e17T + 1.30e35T^{2} \)
53 \( 1 + 1.55e18iT - 1.62e36T^{2} \)
59 \( 1 - 6.38e18T + 1.54e37T^{2} \)
61 \( 1 + 1.92e18iT - 3.10e37T^{2} \)
67 \( 1 - 2.86e18T + 2.22e38T^{2} \)
71 \( 1 + 2.30e19iT - 7.52e38T^{2} \)
73 \( 1 + 1.86e19iT - 1.34e39T^{2} \)
79 \( 1 + 1.57e20T + 7.08e39T^{2} \)
83 \( 1 + 1.76e20T + 1.99e40T^{2} \)
89 \( 1 - 3.24e20T + 8.65e40T^{2} \)
97 \( 1 - 2.37e20iT - 5.27e41T^{2} \)
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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

−13.48459136582672143798680342922, −11.63394853952110494895180162193, −11.27091689331487728568029382736, −8.504201885955545381589550547376, −7.71440121741395505712494310144, −6.72110315514204072583509772695, −5.43231560815140932644129625961, −4.24650589830995719455336213520, −1.77385031898313666277867729474, −0.05362848532188196861118166998, 0.844763450468673189267738562891, 2.55328419649063282471685196384, 4.13830718585795655283499411279, 4.67864798020693091806567479912, 6.93876474053229642111551182990, 8.655832520469403501514162517157, 10.28404693949049249913846484252, 11.20436800127536007362318067435, 11.75845896434638144811116079758, 12.92750250761591034559009550120

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