Properties

Label 2-21-7.2-c5-0-5
Degree $2$
Conductor $21$
Sign $-0.153 + 0.988i$
Analytic cond. $3.36806$
Root an. cond. $1.83522$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4.69 − 8.13i)2-s + (4.5 + 7.79i)3-s + (−28.0 − 48.6i)4-s + (35.8 − 62.1i)5-s + 84.5·6-s + (−87.5 + 95.6i)7-s − 226.·8-s + (−40.5 + 70.1i)9-s + (−336. − 583. i)10-s + (280. + 485. i)11-s + (252. − 437. i)12-s + 533.·13-s + (367. + 1.16e3i)14-s + 645.·15-s + (−166. + 288. i)16-s + (−502. − 870. i)17-s + ⋯
L(s)  = 1  + (0.829 − 1.43i)2-s + (0.288 + 0.499i)3-s + (−0.877 − 1.52i)4-s + (0.641 − 1.11i)5-s + 0.958·6-s + (−0.674 + 0.737i)7-s − 1.25·8-s + (−0.166 + 0.288i)9-s + (−1.06 − 1.84i)10-s + (0.698 + 1.20i)11-s + (0.506 − 0.877i)12-s + 0.875·13-s + (0.500 + 1.58i)14-s + 0.740·15-s + (−0.162 + 0.282i)16-s + (−0.422 − 0.730i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.153 + 0.988i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.153 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $-0.153 + 0.988i$
Analytic conductor: \(3.36806\)
Root analytic conductor: \(1.83522\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :5/2),\ -0.153 + 0.988i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.46487 - 1.70964i\)
\(L(\frac12)\) \(\approx\) \(1.46487 - 1.70964i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-4.5 - 7.79i)T \)
7 \( 1 + (87.5 - 95.6i)T \)
good2 \( 1 + (-4.69 + 8.13i)T + (-16 - 27.7i)T^{2} \)
5 \( 1 + (-35.8 + 62.1i)T + (-1.56e3 - 2.70e3i)T^{2} \)
11 \( 1 + (-280. - 485. i)T + (-8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 - 533.T + 3.71e5T^{2} \)
17 \( 1 + (502. + 870. i)T + (-7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (684. - 1.18e3i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (1.61e3 - 2.79e3i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 + 753.T + 2.05e7T^{2} \)
31 \( 1 + (4.10e3 + 7.10e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + (-1.40e3 + 2.43e3i)T + (-3.46e7 - 6.00e7i)T^{2} \)
41 \( 1 - 245.T + 1.15e8T^{2} \)
43 \( 1 + 1.75e4T + 1.47e8T^{2} \)
47 \( 1 + (-8.17e3 + 1.41e4i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + (-1.48e4 - 2.56e4i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (-5.17e3 - 8.96e3i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (477. - 826. i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (-9.90e3 - 1.71e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 - 6.21e4T + 1.80e9T^{2} \)
73 \( 1 + (1.35e4 + 2.34e4i)T + (-1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (2.23e4 - 3.87e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 - 1.56e4T + 3.93e9T^{2} \)
89 \( 1 + (6.81e3 - 1.18e4i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 + 1.29e4T + 8.58e9T^{2} \)
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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

−16.68722904128151460029649069587, −15.19069639838994768343702892781, −13.66437127125386486412488800915, −12.76463075947576565326796439299, −11.70535350947778750669528392695, −9.889615855458135484476288171625, −9.135784911899209117166342890212, −5.53446207203124009053392215807, −3.99696766055766994802798461055, −1.91163078860558212956076279540, 3.57302174577331376974179687021, 6.26850213285840306833834006994, 6.73206582422545791146051674578, 8.537271723247962785380466805424, 10.74482382656222313337695963124, 13.05348638003073326427462600095, 13.87631549844111119649760677640, 14.61041022410029421707455088785, 16.07019260069426336681165997262, 17.15676954044105743994314177258

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