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Properties

Label 2-21-21.11-c16-0-18
Degree $2$
Conductor $21$
Sign $0.883 + 0.468i$
Analytic cond. $34.0881$
Root an. cond. $5.83850$
Motivic weight $16$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.28e3 + 5.68e3i)3-s + (−3.27e4 + 5.67e4i)4-s + (−5.73e6 + 5.73e5i)7-s + (−2.15e7 − 3.72e7i)9-s + (−2.14e8 − 3.72e8i)12-s − 1.20e9·13-s + (−2.14e9 − 3.71e9i)16-s + (1.19e10 + 2.06e10i)19-s + (1.55e10 − 3.44e10i)21-s + (−7.62e10 + 1.32e11i)25-s + 2.82e11·27-s + (1.55e11 − 3.44e11i)28-s + (1.01e11 − 1.75e11i)31-s + 2.82e12·36-s + (−8.83e11 − 1.52e12i)37-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)3-s + (−0.5 + 0.866i)4-s + (−0.995 + 0.0994i)7-s + (−0.499 − 0.866i)9-s + (−0.5 − 0.866i)12-s − 1.47·13-s + (−0.499 − 0.866i)16-s + (0.702 + 1.21i)19-s + (0.411 − 0.911i)21-s + (−0.5 + 0.866i)25-s + 27-s + (0.411 − 0.911i)28-s + (0.118 − 0.205i)31-s + 0.999·36-s + (−0.251 − 0.435i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.883 + 0.468i)\, \overline{\Lambda}(17-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+8) \, L(s)\cr =\mathstrut & (0.883 + 0.468i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $0.883 + 0.468i$
Analytic conductor: \(34.0881\)
Root analytic conductor: \(5.83850\)
Motivic weight: \(16\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :8),\ 0.883 + 0.468i)\)

Particular Values

\(L(\frac{17}{2})\) \(\approx\) \(0.3121798390\)
\(L(\frac12)\) \(\approx\) \(0.3121798390\)
\(L(9)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (3.28e3 - 5.68e3i)T \)
7 \( 1 + (5.73e6 - 5.73e5i)T \)
good2 \( 1 + (3.27e4 - 5.67e4i)T^{2} \)
5 \( 1 + (7.62e10 - 1.32e11i)T^{2} \)
11 \( 1 + (2.29e16 + 3.97e16i)T^{2} \)
13 \( 1 + 1.20e9T + 6.65e17T^{2} \)
17 \( 1 + (2.43e19 + 4.21e19i)T^{2} \)
19 \( 1 + (-1.19e10 - 2.06e10i)T + (-1.44e20 + 2.49e20i)T^{2} \)
23 \( 1 + (3.06e21 - 5.31e21i)T^{2} \)
29 \( 1 - 2.50e23T^{2} \)
31 \( 1 + (-1.01e11 + 1.75e11i)T + (-3.63e23 - 6.29e23i)T^{2} \)
37 \( 1 + (8.83e11 + 1.52e12i)T + (-6.16e24 + 1.06e25i)T^{2} \)
41 \( 1 - 6.37e25T^{2} \)
43 \( 1 - 2.33e13T + 1.36e26T^{2} \)
47 \( 1 + (2.83e26 - 4.91e26i)T^{2} \)
53 \( 1 + (1.93e27 + 3.35e27i)T^{2} \)
59 \( 1 + (1.07e28 + 1.86e28i)T^{2} \)
61 \( 1 + (9.21e13 + 1.59e14i)T + (-1.83e28 + 3.18e28i)T^{2} \)
67 \( 1 + (2.89e14 - 5.01e14i)T + (-8.24e28 - 1.42e29i)T^{2} \)
71 \( 1 - 4.16e29T^{2} \)
73 \( 1 + (7.19e14 - 1.24e15i)T + (-3.25e29 - 5.63e29i)T^{2} \)
79 \( 1 + (1.28e15 + 2.22e15i)T + (-1.15e30 + 1.99e30i)T^{2} \)
83 \( 1 - 5.07e30T^{2} \)
89 \( 1 + (7.74e30 - 1.34e31i)T^{2} \)
97 \( 1 - 1.56e16T + 6.14e31T^{2} \)
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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

−14.33456021006174895443124797327, −12.72050149621963080405138880711, −11.83676886597536103284025073196, −10.03118491712803341510074065792, −9.187543426116946574710320880828, −7.40559094551865126543128498029, −5.60520130029096884081248828227, −4.13967408207689331256391446081, −2.97968316825045336328788213138, −0.14919050651065554542519296887, 0.77192071892843547020338994618, 2.49034723332367872108629798395, 4.82053383880090644487836296140, 6.11375274074561265591952069944, 7.31807551412554410797416230302, 9.259947904168653561591912381712, 10.44475188677537248769035349558, 12.02861362738924948912521611023, 13.20280165745625455378174840976, 14.20870593026171018778816415169

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