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Properties

Label 2-21-21.20-c5-0-6
Degree $2$
Conductor $21$
Sign $0.994 + 0.101i$
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  + 2.99i·2-s + (−9.94 − 12.0i)3-s + 23.0·4-s + 61.8·5-s + (35.9 − 29.7i)6-s + (92.4 − 90.9i)7-s + 164. i·8-s + (−45.3 + 238. i)9-s + 185. i·10-s − 615. i·11-s + (−228. − 276. i)12-s + 322. i·13-s + (272. + 276. i)14-s + (−614. − 742. i)15-s + 243.·16-s − 1.53e3·17-s + ⋯
L(s)  = 1  + 0.529i·2-s + (−0.637 − 0.770i)3-s + 0.719·4-s + 1.10·5-s + (0.407 − 0.337i)6-s + (0.712 − 0.701i)7-s + 0.910i·8-s + (−0.186 + 0.982i)9-s + 0.585i·10-s − 1.53i·11-s + (−0.459 − 0.554i)12-s + 0.529i·13-s + (0.371 + 0.377i)14-s + (−0.705 − 0.851i)15-s + 0.237·16-s − 1.29·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.101i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.994 + 0.101i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $0.994 + 0.101i$
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: \((2,\ 21,\ (\ :5/2),\ 0.994 + 0.101i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.60787 - 0.0821447i\)
\(L(\frac12)\) \(\approx\) \(1.60787 - 0.0821447i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (9.94 + 12.0i)T \)
7 \( 1 + (-92.4 + 90.9i)T \)
good2 \( 1 - 2.99iT - 32T^{2} \)
5 \( 1 - 61.8T + 3.12e3T^{2} \)
11 \( 1 + 615. iT - 1.61e5T^{2} \)
13 \( 1 - 322. iT - 3.71e5T^{2} \)
17 \( 1 + 1.53e3T + 1.41e6T^{2} \)
19 \( 1 - 2.17e3iT - 2.47e6T^{2} \)
23 \( 1 - 793. iT - 6.43e6T^{2} \)
29 \( 1 + 513. iT - 2.05e7T^{2} \)
31 \( 1 - 161. iT - 2.86e7T^{2} \)
37 \( 1 + 8.59e3T + 6.93e7T^{2} \)
41 \( 1 - 7.00e3T + 1.15e8T^{2} \)
43 \( 1 + 7.72e3T + 1.47e8T^{2} \)
47 \( 1 + 1.76e4T + 2.29e8T^{2} \)
53 \( 1 - 7.34e3iT - 4.18e8T^{2} \)
59 \( 1 - 2.34e4T + 7.14e8T^{2} \)
61 \( 1 - 1.33e4iT - 8.44e8T^{2} \)
67 \( 1 - 1.44e4T + 1.35e9T^{2} \)
71 \( 1 - 7.37e3iT - 1.80e9T^{2} \)
73 \( 1 - 9.42e3iT - 2.07e9T^{2} \)
79 \( 1 - 2.72e4T + 3.07e9T^{2} \)
83 \( 1 - 1.55e4T + 3.93e9T^{2} \)
89 \( 1 - 9.00e4T + 5.58e9T^{2} \)
97 \( 1 - 8.48e4iT - 8.58e9T^{2} \)
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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

−17.05565000310344612337774566039, −16.25943537600008214997243311529, −14.25717974345204505192764500006, −13.47752877809656485569067772263, −11.59327778199653714563747711559, −10.65037590469325691402777249380, −8.220106758602630905503135111210, −6.67466204371716590224649371747, −5.62150791721412454698424784164, −1.74774827895559304788128698119, 2.17950203234029764118238098247, 4.96788068259591657949180281872, 6.61661838917350403555603598488, 9.322711600146374632059197952931, 10.46069146577441226826490155255, 11.58705640347791077396455350352, 12.85654494796650125325861854059, 14.93400207965060474250531863094, 15.70478282177553909723726974764, 17.45731917472995659806595398349

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