Properties

Label 2-21-21.17-c5-0-9
Degree $2$
Conductor $21$
Sign $-0.0156 + 0.999i$
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  + (7.14 − 4.12i)2-s + (−8.42 − 13.1i)3-s + (18.0 − 31.1i)4-s + (−3.57 − 6.18i)5-s + (−114. − 58.8i)6-s + (122.5 − 42.4i)7-s − 32.9i·8-s + (−100. + 221. i)9-s + (−51 − 29.4i)10-s + (389. + 224. i)11-s + (−560. + 26.7i)12-s − 523. i·13-s + (699. − 808. i)14-s + (−50.9 + 98.9i)15-s + (439. + 762. i)16-s + (−592. + 1.02e3i)17-s + ⋯
L(s)  = 1  + (1.26 − 0.728i)2-s + (−0.540 − 0.841i)3-s + (0.562 − 0.974i)4-s + (−0.0638 − 0.110i)5-s + (−1.29 − 0.667i)6-s + (0.944 − 0.327i)7-s − 0.182i·8-s + (−0.415 + 0.909i)9-s + (−0.161 − 0.0931i)10-s + (0.969 + 0.559i)11-s + (−1.12 + 0.0536i)12-s − 0.858i·13-s + (0.954 − 1.10i)14-s + (−0.0585 + 0.113i)15-s + (0.429 + 0.744i)16-s + (−0.497 + 0.861i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0156 + 0.999i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.0156 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $-0.0156 + 0.999i$
Analytic conductor: \(3.36806\)
Root analytic conductor: \(1.83522\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :5/2),\ -0.0156 + 0.999i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.60848 - 1.63386i\)
\(L(\frac12)\) \(\approx\) \(1.60848 - 1.63386i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (8.42 + 13.1i)T \)
7 \( 1 + (-122.5 + 42.4i)T \)
good2 \( 1 + (-7.14 + 4.12i)T + (16 - 27.7i)T^{2} \)
5 \( 1 + (3.57 + 6.18i)T + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (-389. - 224. i)T + (8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 + 523. iT - 3.71e5T^{2} \)
17 \( 1 + (592. - 1.02e3i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (471 - 271. i)T + (1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (3.29e3 - 1.90e3i)T + (3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 + 2.62e3iT - 2.05e7T^{2} \)
31 \( 1 + (7.51e3 + 4.33e3i)T + (1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + (-3.05e3 - 5.28e3i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 + 199.T + 1.15e8T^{2} \)
43 \( 1 + 4.74e3T + 1.47e8T^{2} \)
47 \( 1 + (-1.30e4 - 2.26e4i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (1.19e4 + 6.87e3i)T + (2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (1.45e4 - 2.51e4i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (3.85e3 - 2.22e3i)T + (4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (691 - 1.19e3i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + 8.32e4iT - 1.80e9T^{2} \)
73 \( 1 + (-2.81e4 - 1.62e4i)T + (1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (1.05e4 + 1.82e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 - 3.66e4T + 3.93e9T^{2} \)
89 \( 1 + (3.07e4 + 5.32e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 + 1.20e5iT - 8.58e9T^{2} \)
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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

−17.07813385698061599516745002103, −14.94568025812621349852359941696, −13.88407775893363777938964606385, −12.72033307333396026773846327771, −11.80028000911538675374951070508, −10.72940445672658867391721045390, −7.961824502364955196678881411007, −5.97008250106093372139424855013, −4.34624893800283001571296392263, −1.78117345881310520900385770984, 4.00282045043706396501540201181, 5.26795739969227963958493665748, 6.71378205406842123503667224457, 9.052448337966775250062756264611, 11.13006909321451093930083160594, 12.16768138410973053336396405033, 14.09226847401836089544882179176, 14.73155119348648437117860751430, 15.98047856388217276189476430429, 16.85934139710594784623513425201

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