↑

Properties

Label 2-21-21.20-c5-0-3
Degree $2$
Conductor $21$
Sign $0.0855 - 0.996i$
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

Related objects

Downloads

Learn more

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

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0855 - 0.996i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.0855 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $0.0855 - 0.996i$
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: \((2,\ 21,\ (\ :5/2),\ 0.0855 - 0.996i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.29892 + 1.19217i\)
\(L(\frac12)\) \(\approx\) \(1.29892 + 1.19217i\)
\(L(\frac{7}{2})\) 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 \( 1 + 8.48e4iT - 8.58e9T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−16.92250465888905708697315859353, −15.75568243370242246775373802763, −15.27833380029895034921781935486, −14.05856360413463194600655884529, −11.77386480814045390782350294190, −10.86627789770309310373498126874, −8.613030682939166301762443091446, −7.74934704305209645811751834164, −5.39831175283800354483346163180, −3.14837008572207202202926695387, 1.61138893560570523559857649571, 3.79465271022214912122947768145, 7.10367051124798337112483539758, 7.901710423339336425712311927184, 10.14909997157031681961061765149, 11.82795319654107956178420326956, 12.41138725253646985888029107728, 14.31311481171054108600760317548, 15.29751368078375117406365418119, 16.84786755306955703613410261289

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