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Properties

Label 2-21-21.11-c20-0-23
Degree $2$
Conductor $21$
Sign $0.997 - 0.0738i$
Analytic cond. $53.2378$
Root an. cond. $7.29642$
Motivic weight $20$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.95e4 + 5.11e4i)3-s + (−5.24e5 + 9.08e5i)4-s + (−2.22e8 + 1.73e8i)7-s + (−1.74e9 − 3.01e9i)9-s + (−3.09e10 − 5.36e10i)12-s + 3.81e10·13-s + (−5.49e11 − 9.52e11i)16-s + (−6.00e12 − 1.03e13i)19-s + (−2.28e12 − 1.65e13i)21-s + (−4.76e13 + 8.25e13i)25-s + 2.05e14·27-s + (−4.05e13 − 2.93e14i)28-s + (−4.22e14 + 7.31e14i)31-s + 3.65e15·36-s + (3.60e14 + 6.24e14i)37-s + ⋯
L(s)  = 1  + (−0.499 + 0.866i)3-s + (−0.5 + 0.866i)4-s + (−0.789 + 0.613i)7-s + (−0.5 − 0.866i)9-s + (−0.499 − 0.866i)12-s + 0.276·13-s + (−0.499 − 0.866i)16-s + (−0.979 − 1.69i)19-s + (−0.136 − 0.990i)21-s + (−0.5 + 0.866i)25-s + 27-s + (−0.136 − 0.990i)28-s + (−0.515 + 0.892i)31-s + 0.999·36-s + (0.0749 + 0.129i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0738i)\, \overline{\Lambda}(21-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+10) \, L(s)\cr =\mathstrut & (0.997 - 0.0738i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $0.997 - 0.0738i$
Analytic conductor: \(53.2378\)
Root analytic conductor: \(7.29642\)
Motivic weight: \(20\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :10),\ 0.997 - 0.0738i)\)

Particular Values

\(L(\frac{21}{2})\) \(\approx\) \(0.5754640584\)
\(L(\frac12)\) \(\approx\) \(0.5754640584\)
\(L(11)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.95e4 - 5.11e4i)T \)
7 \( 1 + (2.22e8 - 1.73e8i)T \)
good2 \( 1 + (5.24e5 - 9.08e5i)T^{2} \)
5 \( 1 + (4.76e13 - 8.25e13i)T^{2} \)
11 \( 1 + (3.36e20 + 5.82e20i)T^{2} \)
13 \( 1 - 3.81e10T + 1.90e22T^{2} \)
17 \( 1 + (2.03e24 + 3.51e24i)T^{2} \)
19 \( 1 + (6.00e12 + 1.03e13i)T + (-1.87e25 + 3.25e25i)T^{2} \)
23 \( 1 + (8.58e26 - 1.48e27i)T^{2} \)
29 \( 1 - 1.76e29T^{2} \)
31 \( 1 + (4.22e14 - 7.31e14i)T + (-3.35e29 - 5.81e29i)T^{2} \)
37 \( 1 + (-3.60e14 - 6.24e14i)T + (-1.15e31 + 2.00e31i)T^{2} \)
41 \( 1 - 1.80e32T^{2} \)
43 \( 1 - 1.34e15T + 4.67e32T^{2} \)
47 \( 1 + (1.38e33 - 2.39e33i)T^{2} \)
53 \( 1 + (1.52e34 + 2.64e34i)T^{2} \)
59 \( 1 + (1.30e35 + 2.26e35i)T^{2} \)
61 \( 1 + (-6.93e17 - 1.20e18i)T + (-2.54e35 + 4.40e35i)T^{2} \)
67 \( 1 + (-1.02e18 + 1.78e18i)T + (-1.66e36 - 2.87e36i)T^{2} \)
71 \( 1 - 1.05e37T^{2} \)
73 \( 1 + (-2.38e18 + 4.12e18i)T + (-9.23e36 - 1.59e37i)T^{2} \)
79 \( 1 + (-7.26e18 - 1.25e19i)T + (-4.48e37 + 7.76e37i)T^{2} \)
83 \( 1 - 2.40e38T^{2} \)
89 \( 1 + (4.86e38 - 8.42e38i)T^{2} \)
97 \( 1 + 1.46e20T + 5.43e39T^{2} \)
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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

−13.37190232073269719052296312295, −12.29420958066693377431871922042, −11.06359778754607027220787069768, −9.489054283966129902935339699730, −8.682544840052064427895371799639, −6.74842512205168935321612887885, −5.23315432170616270812467327422, −3.94025732225577878571247253849, −2.80936034111701620158083430381, −0.25564856661877053300332969247, 0.69712901078600067047144790160, 1.94917754515467158211327362728, 4.03888092472912438136834025894, 5.71532758575011323510913237097, 6.57204580922313652210053713294, 8.154622021838432889349669442474, 9.849818151163529075161710952050, 10.87803783619346733736939368996, 12.49393364062664988364948648482, 13.48745082352883059669965236999

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