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Properties

Label 2-21-21.5-c17-0-39
Degree $2$
Conductor $21$
Sign $-0.976 - 0.215i$
Analytic cond. $38.4766$
Root an. cond. $6.20295$
Motivic weight $17$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (9.84e3 − 5.68e3i)3-s + (−6.55e4 − 1.13e5i)4-s + (−1.37e6 + 1.51e7i)7-s + (6.45e7 − 1.11e8i)9-s + (−1.28e9 − 7.44e8i)12-s − 4.89e9i·13-s + (−8.58e9 + 1.48e10i)16-s + (−1.26e11 − 7.33e10i)19-s + (7.27e10 + 1.57e11i)21-s + (3.81e11 + 6.60e11i)25-s − 1.46e12i·27-s + (1.81e12 − 8.38e11i)28-s + (−4.37e12 + 2.52e12i)31-s − 1.69e13·36-s + (−2.10e13 + 3.64e13i)37-s + ⋯
L(s)  = 1  + (0.866 − 0.499i)3-s + (−0.5 − 0.866i)4-s + (−0.0904 + 0.995i)7-s + (0.5 − 0.866i)9-s + (−0.866 − 0.499i)12-s − 1.66i·13-s + (−0.499 + 0.866i)16-s + (−1.71 − 0.990i)19-s + (0.419 + 0.907i)21-s + (0.5 + 0.866i)25-s − 0.999i·27-s + (0.907 − 0.419i)28-s + (−0.920 + 0.531i)31-s − 0.999·36-s + (−0.985 + 1.70i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.976 - 0.215i)\, \overline{\Lambda}(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & (-0.976 - 0.215i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $-0.976 - 0.215i$
Analytic conductor: \(38.4766\)
Root analytic conductor: \(6.20295\)
Motivic weight: \(17\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :17/2),\ -0.976 - 0.215i)\)

Particular Values

\(L(9)\) \(\approx\) \(0.8600789367\)
\(L(\frac12)\) \(\approx\) \(0.8600789367\)
\(L(\frac{19}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-9.84e3 + 5.68e3i)T \)
7 \( 1 + (1.37e6 - 1.51e7i)T \)
good2 \( 1 + (6.55e4 + 1.13e5i)T^{2} \)
5 \( 1 + (-3.81e11 - 6.60e11i)T^{2} \)
11 \( 1 + (2.52e17 - 4.37e17i)T^{2} \)
13 \( 1 + 4.89e9iT - 8.65e18T^{2} \)
17 \( 1 + (-4.13e20 + 7.16e20i)T^{2} \)
19 \( 1 + (1.26e11 + 7.33e10i)T + (2.74e21 + 4.74e21i)T^{2} \)
23 \( 1 + (7.05e22 + 1.22e23i)T^{2} \)
29 \( 1 - 7.25e24T^{2} \)
31 \( 1 + (4.37e12 - 2.52e12i)T + (1.12e25 - 1.95e25i)T^{2} \)
37 \( 1 + (2.10e13 - 3.64e13i)T + (-2.28e26 - 3.95e26i)T^{2} \)
41 \( 1 + 2.61e27T^{2} \)
43 \( 1 + 5.48e13T + 5.87e27T^{2} \)
47 \( 1 + (-1.33e28 - 2.30e28i)T^{2} \)
53 \( 1 + (1.02e29 - 1.77e29i)T^{2} \)
59 \( 1 + (-6.35e29 + 1.10e30i)T^{2} \)
61 \( 1 + (2.59e15 + 1.49e15i)T + (1.12e30 + 1.94e30i)T^{2} \)
67 \( 1 + (-1.34e14 - 2.33e14i)T + (-5.52e30 + 9.56e30i)T^{2} \)
71 \( 1 - 2.96e31T^{2} \)
73 \( 1 + (-7.49e15 + 4.32e15i)T + (2.37e31 - 4.11e31i)T^{2} \)
79 \( 1 + (-1.32e16 + 2.29e16i)T + (-9.09e31 - 1.57e32i)T^{2} \)
83 \( 1 + 4.21e32T^{2} \)
89 \( 1 + (-6.89e32 - 1.19e33i)T^{2} \)
97 \( 1 - 9.91e16iT - 5.95e33T^{2} \)
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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

−13.43189344608350551498973227791, −12.58127377865421462491688527886, −10.62002563956613514908130527719, −9.212154440367386356237567623329, −8.311901461894302737167807047891, −6.47156239331448620788643065646, −5.06743510926847890243346793662, −3.09718009268942359849182785808, −1.71783164558578473133297397705, −0.21030744653775516846655122657, 2.04859868907675581650995899585, 3.77792064457441427014083380539, 4.37742455486164328761981449108, 6.99244850594775641708476934736, 8.276911895899791172137384170713, 9.334461195804108735254837475357, 10.72987156030919159941267717933, 12.52906423313397648282384532445, 13.74031202303732162307301646737, 14.52442582796849976908594628050

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