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Properties

Label 2-21-21.17-c31-0-22
Degree $2$
Conductor $21$
Sign $-0.869 + 0.493i$
Analytic cond. $127.841$
Root an. cond. $11.3067$
Motivic weight $31$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.15e7 + 1.24e7i)3-s + (−1.07e9 + 1.85e9i)4-s + (4.76e12 + 1.16e13i)7-s + (3.08e14 + 5.34e14i)9-s + (−4.62e16 + 2.66e16i)12-s + 3.48e17i·13-s + (−2.30e18 − 3.99e18i)16-s + (−9.29e19 + 5.36e19i)19-s + (−4.18e19 + 3.09e20i)21-s + (2.32e21 − 4.03e21i)25-s + (−1.04e6 + 1.53e22i)27-s + (−2.67e22 − 3.61e21i)28-s + (1.37e23 + 7.93e22i)31-s + (−1.32e24 − 6.71e7i)36-s + (1.12e24 + 1.95e24i)37-s + ⋯
L(s)  = 1  + (0.866 + 0.499i)3-s + (−0.5 + 0.866i)4-s + (0.379 + 0.925i)7-s + (0.499 + 0.866i)9-s + (−0.866 + 0.499i)12-s + 1.88i·13-s + (−0.499 − 0.866i)16-s + (−1.40 + 0.810i)19-s + (−0.133 + 0.990i)21-s + (0.5 − 0.866i)25-s + 1.00i·27-s + (−0.990 − 0.133i)28-s + (1.05 + 0.607i)31-s − 0.999·36-s + (0.556 + 0.964i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.869 + 0.493i)\, \overline{\Lambda}(32-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+31/2) \, L(s)\cr =\mathstrut & (-0.869 + 0.493i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $-0.869 + 0.493i$
Analytic conductor: \(127.841\)
Root analytic conductor: \(11.3067\)
Motivic weight: \(31\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :31/2),\ -0.869 + 0.493i)\)

Particular Values

\(L(16)\) \(\approx\) \(2.183434824\)
\(L(\frac12)\) \(\approx\) \(2.183434824\)
\(L(\frac{33}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.15e7 - 1.24e7i)T \)
7 \( 1 + (-4.76e12 - 1.16e13i)T \)
good2 \( 1 + (1.07e9 - 1.85e9i)T^{2} \)
5 \( 1 + (-2.32e21 + 4.03e21i)T^{2} \)
11 \( 1 + (9.59e31 + 1.66e32i)T^{2} \)
13 \( 1 - 3.48e17iT - 3.40e34T^{2} \)
17 \( 1 + (-6.96e37 - 1.20e38i)T^{2} \)
19 \( 1 + (9.29e19 - 5.36e19i)T + (2.18e39 - 3.79e39i)T^{2} \)
23 \( 1 + (8.17e41 - 1.41e42i)T^{2} \)
29 \( 1 - 2.15e45T^{2} \)
31 \( 1 + (-1.37e23 - 7.93e22i)T + (8.53e45 + 1.47e46i)T^{2} \)
37 \( 1 + (-1.12e24 - 1.95e24i)T + (-2.05e48 + 3.56e48i)T^{2} \)
41 \( 1 + 9.91e49T^{2} \)
43 \( 1 + 1.77e25T + 4.34e50T^{2} \)
47 \( 1 + (-3.41e51 + 5.92e51i)T^{2} \)
53 \( 1 + (1.41e53 + 2.45e53i)T^{2} \)
59 \( 1 + (-3.93e54 - 6.82e54i)T^{2} \)
61 \( 1 + (-8.10e27 + 4.67e27i)T + (1.10e55 - 1.91e55i)T^{2} \)
67 \( 1 + (1.51e28 - 2.62e28i)T + (-2.02e56 - 3.51e56i)T^{2} \)
71 \( 1 - 2.44e57T^{2} \)
73 \( 1 + (-1.14e29 - 6.61e28i)T + (2.89e57 + 5.01e57i)T^{2} \)
79 \( 1 + (7.51e28 + 1.30e29i)T + (-3.35e58 + 5.80e58i)T^{2} \)
83 \( 1 + 3.10e59T^{2} \)
89 \( 1 + (-1.34e60 + 2.33e60i)T^{2} \)
97 \( 1 + 1.01e31iT - 3.88e61T^{2} \)
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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

−12.62353885978912328473551737717, −11.54808025042274190129700650190, −9.857618411194441065726705714071, −8.709538621993976808827741191590, −8.277833599662984151333247277969, −6.66538639942885067327519578196, −4.75573230145445970656397846709, −4.05193767436502338981831546343, −2.72871058213817004823156770962, −1.81205679494473463239036725342, 0.42522760492814793858665364861, 1.05335691895801004512252083791, 2.37603376448651128012558427420, 3.75141612517288455401588092197, 4.95784978886825583717247975933, 6.39666040115453478095745301280, 7.69294964111066011318568505262, 8.687569246953655419150345543546, 9.987914325070084230725721168640, 10.89480972929384660825402227617

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