Properties

Label 2-33-3.2-c2-0-0
Degree $2$
Conductor $33$
Sign $-0.445 - 0.895i$
Analytic cond. $0.899184$
Root an. cond. $0.948253$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2.52i·2-s + (−2.68 + 1.33i)3-s − 2.37·4-s + 0.792i·5-s + (−3.37 − 6.78i)6-s + 6.74·7-s + 4.10i·8-s + (5.43 − 7.17i)9-s − 2·10-s − 3.31i·11-s + (6.37 − 3.16i)12-s + 9.48·13-s + 17.0i·14-s + (−1.05 − 2.12i)15-s − 19.8·16-s − 29.2i·17-s + ⋯
L(s)  = 1  + 1.26i·2-s + (−0.895 + 0.445i)3-s − 0.593·4-s + 0.158i·5-s + (−0.562 − 1.13i)6-s + 0.963·7-s + 0.513i·8-s + (0.603 − 0.797i)9-s − 0.200·10-s − 0.301i·11-s + (0.531 − 0.264i)12-s + 0.729·13-s + 1.21i·14-s + (−0.0705 − 0.141i)15-s − 1.24·16-s − 1.72i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.445 - 0.895i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.445 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $-0.445 - 0.895i$
Analytic conductor: \(0.899184\)
Root analytic conductor: \(0.948253\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{33} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 33,\ (\ :1),\ -0.445 - 0.895i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.463658 + 0.748427i\)
\(L(\frac12)\) \(\approx\) \(0.463658 + 0.748427i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.68 - 1.33i)T \)
11 \( 1 + 3.31iT \)
good2 \( 1 - 2.52iT - 4T^{2} \)
5 \( 1 - 0.792iT - 25T^{2} \)
7 \( 1 - 6.74T + 49T^{2} \)
13 \( 1 - 9.48T + 169T^{2} \)
17 \( 1 + 29.2iT - 289T^{2} \)
19 \( 1 + 26.2T + 361T^{2} \)
23 \( 1 - 26.9iT - 529T^{2} \)
29 \( 1 + 25.9iT - 841T^{2} \)
31 \( 1 + 2.86T + 961T^{2} \)
37 \( 1 + 2.39T + 1.36e3T^{2} \)
41 \( 1 + 17.6iT - 1.68e3T^{2} \)
43 \( 1 - 12.5T + 1.84e3T^{2} \)
47 \( 1 + 41.6iT - 2.20e3T^{2} \)
53 \( 1 - 89.5iT - 2.80e3T^{2} \)
59 \( 1 - 14.7iT - 3.48e3T^{2} \)
61 \( 1 + 63.4T + 3.72e3T^{2} \)
67 \( 1 + 63.3T + 4.48e3T^{2} \)
71 \( 1 + 4.55iT - 5.04e3T^{2} \)
73 \( 1 + 53.7T + 5.32e3T^{2} \)
79 \( 1 - 55.6T + 6.24e3T^{2} \)
83 \( 1 + 65.3iT - 6.88e3T^{2} \)
89 \( 1 + 14.1iT - 7.92e3T^{2} \)
97 \( 1 - 149.T + 9.40e3T^{2} \)
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

−16.76417928530706413751207276223, −15.78590919128044371058837590570, −14.92712728876636484495397950383, −13.69327895828834475894436114817, −11.70968852981559669781787698033, −10.84036526998441555175128507846, −8.918944740889346157887122516605, −7.37441175036494008532991693648, −6.02035549250298108280615147416, −4.76262208178670543919733466368, 1.65654041489718236067155511163, 4.45329971210931960882440347768, 6.46829480048815188021083183211, 8.429983005306895403306275961389, 10.56179720247796994474597070107, 11.00486444852050747712896245855, 12.38141397789172462770821812042, 12.99422447532934789555777570233, 14.79763435069408702750368823623, 16.43020260714070259938855108752

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