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Properties

Label 2-21-7.4-c5-0-6
Degree $2$
Conductor $21$
Sign $-0.970 + 0.240i$
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  + (−3.19 − 5.53i)2-s + (4.5 − 7.79i)3-s + (−4.41 + 7.64i)4-s + (−19.3 − 33.5i)5-s − 57.5·6-s + (−87.5 + 95.6i)7-s − 148.·8-s + (−40.5 − 70.1i)9-s + (−123. + 214. i)10-s + (288. − 499. i)11-s + (39.7 + 68.8i)12-s + 391.·13-s + (808. + 178. i)14-s − 348.·15-s + (614. + 1.06e3i)16-s + (664. − 1.15e3i)17-s + ⋯
L(s)  = 1  + (−0.564 − 0.978i)2-s + (0.288 − 0.499i)3-s + (−0.137 + 0.238i)4-s + (−0.346 − 0.599i)5-s − 0.652·6-s + (−0.674 + 0.737i)7-s − 0.817·8-s + (−0.166 − 0.288i)9-s + (−0.391 + 0.677i)10-s + (0.718 − 1.24i)11-s + (0.0796 + 0.137i)12-s + 0.642·13-s + (1.10 + 0.243i)14-s − 0.399·15-s + (0.599 + 1.03i)16-s + (0.557 − 0.966i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.970 + 0.240i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.970 + 0.240i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $-0.970 + 0.240i$
Analytic conductor: \(3.36806\)
Root analytic conductor: \(1.83522\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :5/2),\ -0.970 + 0.240i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.114791 - 0.940356i\)
\(L(\frac12)\) \(\approx\) \(0.114791 - 0.940356i\)
\(L(\frac{7}{2})\) 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 \( 1 + (3.19 + 5.53i)T + (-16 + 27.7i)T^{2} \)
5 \( 1 + (19.3 + 33.5i)T + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (-288. + 499. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 - 391.T + 3.71e5T^{2} \)
17 \( 1 + (-664. + 1.15e3i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (471. + 816. i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (-816. - 1.41e3i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + 1.46e3T + 2.05e7T^{2} \)
31 \( 1 + (-1.95e3 + 3.38e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (-8.15e3 - 1.41e4i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 + 1.31e4T + 1.15e8T^{2} \)
43 \( 1 - 1.47e4T + 1.47e8T^{2} \)
47 \( 1 + (-3.40e3 - 5.90e3i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (-1.00e3 + 1.74e3i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (2.57e4 - 4.45e4i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (2.05e4 + 3.55e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (2.52e4 - 4.38e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 - 3.99e4T + 1.80e9T^{2} \)
73 \( 1 + (-2.78e4 + 4.82e4i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (-3.15e4 - 5.46e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 - 4.55e4T + 3.93e9T^{2} \)
89 \( 1 + (7.84e3 + 1.35e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 - 3.12e3T + 8.58e9T^{2} \)
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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

−16.66509318585605293816148855240, −15.34713987950776964078557677214, −13.57297615398159351860169888367, −12.21420001837697700783114851422, −11.32754009764430338046869657439, −9.429180771353673380465630164394, −8.526987354037956290417673156129, −6.14476899852555871958750668652, −3.05949196601917328195645533906, −0.847065099873545780364881394854, 3.70396416083621652751346224856, 6.47103772572418322098874192107, 7.65268311827782394120043657193, 9.245271005216166249143835702644, 10.58961778243552219371916689167, 12.50507141260342186601110567731, 14.45488707215421780811473907423, 15.28091175938255730592532770559, 16.46621334729973753839585077466, 17.32457896799792680685549169209

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