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Properties

Label 2-21-21.17-c41-0-69
Degree $2$
Conductor $21$
Sign $-0.0678 - 0.997i$
Analytic cond. $223.590$
Root an. cond. $14.9529$
Motivic weight $41$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (5.23e9 + 3.01e9i)3-s + (−1.09e12 + 1.90e12i)4-s + (2.10e17 + 1.24e16i)7-s + (1.82e19 + 3.15e19i)9-s + (−1.15e22 + 6.64e21i)12-s + 6.81e22i·13-s + (−2.41e24 − 4.18e24i)16-s + (2.45e26 − 1.41e26i)19-s + (1.06e27 + 7.01e26i)21-s + (2.27e28 − 3.93e28i)25-s + (1.75e13 + 2.20e29i)27-s + (−2.55e29 + 3.87e29i)28-s + (5.95e30 + 3.43e30i)31-s − 8.02e31·36-s + (3.23e31 + 5.61e31i)37-s + ⋯
L(s)  = 1  + (0.866 + 0.499i)3-s + (−0.5 + 0.866i)4-s + (0.998 + 0.0587i)7-s + (0.5 + 0.866i)9-s + (−0.866 + 0.499i)12-s + 0.994i·13-s + (−0.499 − 0.866i)16-s + (1.50 − 0.866i)19-s + (0.835 + 0.550i)21-s + (0.5 − 0.866i)25-s + 0.999i·27-s + (−0.550 + 0.835i)28-s + (1.59 + 0.919i)31-s − 36-s + (0.230 + 0.398i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0678 - 0.997i)\, \overline{\Lambda}(42-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+41/2) \, L(s)\cr =\mathstrut & (-0.0678 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $-0.0678 - 0.997i$
Analytic conductor: \(223.590\)
Root analytic conductor: \(14.9529\)
Motivic weight: \(41\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :41/2),\ -0.0678 - 0.997i)\)

Particular Values

\(L(21)\) \(\approx\) \(3.995299661\)
\(L(\frac12)\) \(\approx\) \(3.995299661\)
\(L(\frac{43}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-5.23e9 - 3.01e9i)T \)
7 \( 1 + (-2.10e17 - 1.24e16i)T \)
good2 \( 1 + (1.09e12 - 1.90e12i)T^{2} \)
5 \( 1 + (-2.27e28 + 3.93e28i)T^{2} \)
11 \( 1 + (2.48e42 + 4.31e42i)T^{2} \)
13 \( 1 - 6.81e22iT - 4.69e45T^{2} \)
17 \( 1 + (-1.40e50 - 2.43e50i)T^{2} \)
19 \( 1 + (-2.45e26 + 1.41e26i)T + (1.34e52 - 2.32e52i)T^{2} \)
23 \( 1 + (3.38e55 - 5.86e55i)T^{2} \)
29 \( 1 - 9.08e59T^{2} \)
31 \( 1 + (-5.95e30 - 3.43e30i)T + (6.99e60 + 1.21e61i)T^{2} \)
37 \( 1 + (-3.23e31 - 5.61e31i)T + (-9.89e63 + 1.71e64i)T^{2} \)
41 \( 1 + 1.33e66T^{2} \)
43 \( 1 - 1.97e33T + 9.38e66T^{2} \)
47 \( 1 + (-1.79e68 + 3.11e68i)T^{2} \)
53 \( 1 + (2.47e70 + 4.29e70i)T^{2} \)
59 \( 1 + (-2.01e72 - 3.48e72i)T^{2} \)
61 \( 1 + (-5.89e34 + 3.40e34i)T + (7.89e72 - 1.36e73i)T^{2} \)
67 \( 1 + (-2.54e37 + 4.40e37i)T + (-3.69e74 - 6.40e74i)T^{2} \)
71 \( 1 - 7.97e75T^{2} \)
73 \( 1 + (-7.06e36 - 4.07e36i)T + (1.24e76 + 2.15e76i)T^{2} \)
79 \( 1 + (-6.37e38 - 1.10e39i)T + (-3.17e77 + 5.49e77i)T^{2} \)
83 \( 1 + 4.81e78T^{2} \)
89 \( 1 + (-4.20e79 + 7.28e79i)T^{2} \)
97 \( 1 + 5.81e40iT - 2.86e81T^{2} \)
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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

−11.21223140107970787120981735834, −9.729557103018931694900227227480, −8.792004070059424369435078433936, −8.038774479900547269829015890780, −7.02468607768345997735193278197, −4.94168482755603256248106915048, −4.40510669530501219560262078576, −3.21721560019215012738844096223, −2.31777806108643967231617207371, −0.990598765625308840292502345926, 0.793124078707899637314877223205, 1.27117362864011335639735175913, 2.45535495047263783072128051453, 3.72143270098241994231105772719, 4.95593714403836570156209039707, 5.94707336058419198824362251194, 7.46653929108497972248748988285, 8.268597323614005815220727621696, 9.391568685295201280921442882501, 10.35913235546043499243776371673

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