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Properties

Label 2-21-21.20-c29-0-30
Degree $2$
Conductor $21$
Sign $-0.955 + 0.293i$
Analytic cond. $111.883$
Root an. cond. $10.5775$
Motivic weight $29$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.05e4i·2-s + (−8.96e5 − 8.23e6i)3-s − 1.10e9·4-s + 2.53e10·5-s + (3.33e11 − 3.63e10i)6-s + (3.38e11 + 1.76e12i)7-s − 2.30e13i·8-s + (−6.70e13 + 1.47e13i)9-s + 1.02e15i·10-s + 1.41e15i·11-s + (9.91e14 + 9.11e15i)12-s + 2.20e16i·13-s + (−7.14e16 + 1.37e16i)14-s + (−2.27e16 − 2.08e17i)15-s + 3.41e17·16-s + 9.03e17·17-s + ⋯
L(s)  = 1  + 1.74i·2-s + (−0.108 − 0.994i)3-s − 2.06·4-s + 1.85·5-s + (1.73 − 0.189i)6-s + (0.188 + 0.982i)7-s − 1.85i·8-s + (−0.976 + 0.215i)9-s + 3.24i·10-s + 1.12i·11-s + (0.222 + 2.04i)12-s + 1.55i·13-s + (−1.71 + 0.330i)14-s + (−0.200 − 1.84i)15-s + 1.18·16-s + 1.30·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.955 + 0.293i)\, \overline{\Lambda}(30-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+29/2) \, L(s)\cr =\mathstrut & (-0.955 + 0.293i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $-0.955 + 0.293i$
Analytic conductor: \(111.883\)
Root analytic conductor: \(10.5775\)
Motivic weight: \(29\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :29/2),\ -0.955 + 0.293i)\)

Particular Values

\(L(15)\) \(\approx\) \(2.508848351\)
\(L(\frac12)\) \(\approx\) \(2.508848351\)
\(L(\frac{31}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (8.96e5 + 8.23e6i)T \)
7 \( 1 + (-3.38e11 - 1.76e12i)T \)
good2 \( 1 - 4.05e4iT - 5.36e8T^{2} \)
5 \( 1 - 2.53e10T + 1.86e20T^{2} \)
11 \( 1 - 1.41e15iT - 1.58e30T^{2} \)
13 \( 1 - 2.20e16iT - 2.01e32T^{2} \)
17 \( 1 - 9.03e17T + 4.81e35T^{2} \)
19 \( 1 + 2.84e17iT - 1.21e37T^{2} \)
23 \( 1 - 3.67e19iT - 3.09e39T^{2} \)
29 \( 1 + 1.16e21iT - 2.56e42T^{2} \)
31 \( 1 + 1.18e21iT - 1.77e43T^{2} \)
37 \( 1 + 1.35e21T + 3.00e45T^{2} \)
41 \( 1 - 1.70e23T + 5.89e46T^{2} \)
43 \( 1 + 6.75e23T + 2.34e47T^{2} \)
47 \( 1 - 1.96e24T + 3.09e48T^{2} \)
53 \( 1 + 1.77e24iT - 1.00e50T^{2} \)
59 \( 1 + 4.83e25T + 2.26e51T^{2} \)
61 \( 1 - 7.56e25iT - 5.95e51T^{2} \)
67 \( 1 + 1.04e26T + 9.04e52T^{2} \)
71 \( 1 + 2.52e26iT - 4.85e53T^{2} \)
73 \( 1 + 9.99e26iT - 1.08e54T^{2} \)
79 \( 1 + 2.17e27T + 1.07e55T^{2} \)
83 \( 1 - 6.73e27T + 4.50e55T^{2} \)
89 \( 1 + 8.13e27T + 3.40e56T^{2} \)
97 \( 1 - 1.01e29iT - 4.13e57T^{2} \)
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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

−13.33107778781961936286602292696, −12.08608955956110778777900022229, −9.654124086729696280523046780106, −8.939632925491922670607186442070, −7.52002719310077310703047432356, −6.47108515614676278656370607984, −5.80601399385175940109868860668, −4.94423465129695056507137339341, −2.29377855102041679510682349709, −1.45493141109941068200206979771, 0.54394737681124038589375246223, 1.30650724571061308756925341495, 2.79534239548289912770014411113, 3.44281138553254344466636063132, 4.97081131841682174131276925231, 5.82868490325931630605108814489, 8.541905448983322894299262364787, 9.762437931217028593322860207598, 10.37159084455447013335921776467, 10.91866215334184647559533108545

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