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Properties

Label 2-21-21.20-c5-0-0
Degree $2$
Conductor $21$
Sign $-0.735 + 0.677i$
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  + 10.2i·2-s + (−4.60 − 14.8i)3-s − 72.9·4-s − 73.2·5-s + (152. − 47.1i)6-s + (55.7 + 117. i)7-s − 419. i·8-s + (−200. + 137. i)9-s − 751. i·10-s + 237. i·11-s + (335. + 1.08e3i)12-s + 63.3i·13-s + (−1.19e3 + 570. i)14-s + (337. + 1.09e3i)15-s + 1.96e3·16-s − 814.·17-s + ⋯
L(s)  = 1  + 1.81i·2-s + (−0.295 − 0.955i)3-s − 2.28·4-s − 1.31·5-s + (1.73 − 0.534i)6-s + (0.429 + 0.902i)7-s − 2.31i·8-s + (−0.825 + 0.563i)9-s − 2.37i·10-s + 0.593i·11-s + (0.672 + 2.17i)12-s + 0.104i·13-s + (−1.63 + 0.778i)14-s + (0.386 + 1.25i)15-s + 1.92·16-s − 0.683·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.735 + 0.677i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.735 + 0.677i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $-0.735 + 0.677i$
Analytic conductor: \(3.36806\)
Root analytic conductor: \(1.83522\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :5/2),\ -0.735 + 0.677i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.152483 - 0.390889i\)
\(L(\frac12)\) \(\approx\) \(0.152483 - 0.390889i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (4.60 + 14.8i)T \)
7 \( 1 + (-55.7 - 117. i)T \)
good2 \( 1 - 10.2iT - 32T^{2} \)
5 \( 1 + 73.2T + 3.12e3T^{2} \)
11 \( 1 - 237. iT - 1.61e5T^{2} \)
13 \( 1 - 63.3iT - 3.71e5T^{2} \)
17 \( 1 + 814.T + 1.41e6T^{2} \)
19 \( 1 + 64.1iT - 2.47e6T^{2} \)
23 \( 1 + 988. iT - 6.43e6T^{2} \)
29 \( 1 - 1.56e3iT - 2.05e7T^{2} \)
31 \( 1 - 5.25e3iT - 2.86e7T^{2} \)
37 \( 1 - 1.13e4T + 6.93e7T^{2} \)
41 \( 1 + 3.79e3T + 1.15e8T^{2} \)
43 \( 1 + 1.58e4T + 1.47e8T^{2} \)
47 \( 1 + 2.32e4T + 2.29e8T^{2} \)
53 \( 1 - 1.99e4iT - 4.18e8T^{2} \)
59 \( 1 - 3.45e4T + 7.14e8T^{2} \)
61 \( 1 + 1.20e4iT - 8.44e8T^{2} \)
67 \( 1 + 2.92e4T + 1.35e9T^{2} \)
71 \( 1 + 5.40e4iT - 1.80e9T^{2} \)
73 \( 1 - 7.37e4iT - 2.07e9T^{2} \)
79 \( 1 + 4.92e4T + 3.07e9T^{2} \)
83 \( 1 - 1.12e5T + 3.93e9T^{2} \)
89 \( 1 + 7.17e4T + 5.58e9T^{2} \)
97 \( 1 - 1.80e4iT - 8.58e9T^{2} \)
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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

−17.80867869491232976289299855080, −16.47439157538131311104346177577, −15.40440706992517930367702634491, −14.51407389452783859441173751898, −12.91415902336027160621675677624, −11.65516506583917971550589833774, −8.678995645812360760464167281540, −7.75897159697716687383492350468, −6.57480302820617978215271775035, −4.88855566014675110696910846684, 0.32346105444179861885746270660, 3.53722294388708886316723898460, 4.53618401521729189447121888764, 8.331611061619992494353204727346, 9.945494723784323977674173631636, 11.21893210941193071543090582937, 11.58100329890728388092072322248, 13.34239177321392397896799045844, 14.89112267839370288778102595008, 16.46504554182653009012616001977

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