↑

Properties

Label 2-33-3.2-c6-0-2
Degree $2$
Conductor $33$
Sign $0.230 + 0.973i$
Analytic cond. $7.59178$
Root an. cond. $2.75531$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 14.5i·2-s + (−26.2 + 6.21i)3-s − 147.·4-s + 220. i·5-s + (−90.4 − 381. i)6-s + 126.·7-s − 1.21e3i·8-s + (651. − 326. i)9-s − 3.20e3·10-s + 401. i·11-s + (3.87e3 − 916. i)12-s + 1.70e3·13-s + 1.83e3i·14-s + (−1.37e3 − 5.80e3i)15-s + 8.18e3·16-s + 5.01e3i·17-s + ⋯
L(s)  = 1  + 1.81i·2-s + (−0.973 + 0.230i)3-s − 2.30·4-s + 1.76i·5-s + (−0.418 − 1.76i)6-s + 0.367·7-s − 2.36i·8-s + (0.893 − 0.448i)9-s − 3.20·10-s + 0.301i·11-s + (2.24 − 0.530i)12-s + 0.777·13-s + 0.668i·14-s + (−0.406 − 1.71i)15-s + 1.99·16-s + 1.02i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.230 + 0.973i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.230 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $0.230 + 0.973i$
Analytic conductor: \(7.59178\)
Root analytic conductor: \(2.75531\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{33} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 33,\ (\ :3),\ 0.230 + 0.973i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.584130 - 0.462015i\)
\(L(\frac12)\) \(\approx\) \(0.584130 - 0.462015i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (26.2 - 6.21i)T \)
11 \( 1 - 401. iT \)
good2 \( 1 - 14.5iT - 64T^{2} \)
5 \( 1 - 220. iT - 1.56e4T^{2} \)
7 \( 1 - 126.T + 1.17e5T^{2} \)
13 \( 1 - 1.70e3T + 4.82e6T^{2} \)
17 \( 1 - 5.01e3iT - 2.41e7T^{2} \)
19 \( 1 + 6.95e3T + 4.70e7T^{2} \)
23 \( 1 + 1.09e4iT - 1.48e8T^{2} \)
29 \( 1 + 1.26e4iT - 5.94e8T^{2} \)
31 \( 1 - 9.40e3T + 8.87e8T^{2} \)
37 \( 1 - 8.56e4T + 2.56e9T^{2} \)
41 \( 1 - 1.03e5iT - 4.75e9T^{2} \)
43 \( 1 + 7.51e4T + 6.32e9T^{2} \)
47 \( 1 - 1.43e4iT - 1.07e10T^{2} \)
53 \( 1 + 1.58e4iT - 2.21e10T^{2} \)
59 \( 1 - 8.48e4iT - 4.21e10T^{2} \)
61 \( 1 - 2.01e5T + 5.15e10T^{2} \)
67 \( 1 - 1.23e5T + 9.04e10T^{2} \)
71 \( 1 - 3.91e4iT - 1.28e11T^{2} \)
73 \( 1 + 5.54e5T + 1.51e11T^{2} \)
79 \( 1 - 8.61e4T + 2.43e11T^{2} \)
83 \( 1 + 4.95e5iT - 3.26e11T^{2} \)
89 \( 1 - 6.93e5iT - 4.96e11T^{2} \)
97 \( 1 + 1.46e6T + 8.32e11T^{2} \)
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

−16.29921679354097402847045988064, −15.03948067227225561378718277868, −14.73156233772539587712704135205, −13.16692082013584408641046164663, −11.17023629359604241723726998222, −10.04404312390337790109672307427, −8.055688911006708231934742341074, −6.64523963477703783733697956059, −6.11312040006404385869506184945, −4.25226926059896191633209034936, 0.48222581655028795523124252567, 1.55154389967018886631601031849, 4.27619222102705615137722128311, 5.35197498527519476167051263178, 8.429693845809753859167116667617, 9.612804317407401332005293815233, 11.08517903399458253111713012709, 11.89175772415914355162406771555, 12.85843138864720544682113671962, 13.55101161620218691966885954222

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