Properties

Label 2-33-11.2-c2-0-3
Degree $2$
Conductor $33$
Sign $0.999 - 0.0136i$
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  + (1.28 + 0.419i)2-s + (1.40 − 1.01i)3-s + (−1.74 − 1.27i)4-s + (0.708 + 2.18i)5-s + (2.23 − 0.725i)6-s + (−5.74 + 7.90i)7-s + (−4.91 − 6.75i)8-s + (0.927 − 2.85i)9-s + 3.11i·10-s + (10.3 − 3.76i)11-s − 3.74·12-s + (−14.6 − 4.75i)13-s + (−10.7 + 7.78i)14-s + (3.21 + 2.33i)15-s + (−0.830 − 2.55i)16-s + (11.2 − 3.65i)17-s + ⋯
L(s)  = 1  + (0.644 + 0.209i)2-s + (0.467 − 0.339i)3-s + (−0.437 − 0.317i)4-s + (0.141 + 0.436i)5-s + (0.372 − 0.120i)6-s + (−0.820 + 1.12i)7-s + (−0.613 − 0.844i)8-s + (0.103 − 0.317i)9-s + 0.311i·10-s + (0.939 − 0.342i)11-s − 0.311·12-s + (−1.12 − 0.365i)13-s + (−0.765 + 0.556i)14-s + (0.214 + 0.155i)15-s + (−0.0519 − 0.159i)16-s + (0.661 − 0.214i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0136i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 - 0.0136i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $0.999 - 0.0136i$
Analytic conductor: \(0.899184\)
Root analytic conductor: \(0.948253\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{33} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 33,\ (\ :1),\ 0.999 - 0.0136i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.30735 + 0.00890925i\)
\(L(\frac12)\) \(\approx\) \(1.30735 + 0.00890925i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.40 + 1.01i)T \)
11 \( 1 + (-10.3 + 3.76i)T \)
good2 \( 1 + (-1.28 - 0.419i)T + (3.23 + 2.35i)T^{2} \)
5 \( 1 + (-0.708 - 2.18i)T + (-20.2 + 14.6i)T^{2} \)
7 \( 1 + (5.74 - 7.90i)T + (-15.1 - 46.6i)T^{2} \)
13 \( 1 + (14.6 + 4.75i)T + (136. + 99.3i)T^{2} \)
17 \( 1 + (-11.2 + 3.65i)T + (233. - 169. i)T^{2} \)
19 \( 1 + (-7.10 - 9.77i)T + (-111. + 343. i)T^{2} \)
23 \( 1 - 16.6T + 529T^{2} \)
29 \( 1 + (15.6 - 21.4i)T + (-259. - 799. i)T^{2} \)
31 \( 1 + (-1.28 + 3.95i)T + (-777. - 564. i)T^{2} \)
37 \( 1 + (54.4 + 39.5i)T + (423. + 1.30e3i)T^{2} \)
41 \( 1 + (10.6 + 14.6i)T + (-519. + 1.59e3i)T^{2} \)
43 \( 1 - 46.3iT - 1.84e3T^{2} \)
47 \( 1 + (49.2 - 35.8i)T + (682. - 2.10e3i)T^{2} \)
53 \( 1 + (-31.2 + 96.2i)T + (-2.27e3 - 1.65e3i)T^{2} \)
59 \( 1 + (-78.7 - 57.1i)T + (1.07e3 + 3.31e3i)T^{2} \)
61 \( 1 + (-1.19 + 0.388i)T + (3.01e3 - 2.18e3i)T^{2} \)
67 \( 1 + 55.1T + 4.48e3T^{2} \)
71 \( 1 + (4.62 + 14.2i)T + (-4.07e3 + 2.96e3i)T^{2} \)
73 \( 1 + (44.6 - 61.3i)T + (-1.64e3 - 5.06e3i)T^{2} \)
79 \( 1 + (-27.7 - 9.02i)T + (5.04e3 + 3.66e3i)T^{2} \)
83 \( 1 + (6.06 - 1.96i)T + (5.57e3 - 4.04e3i)T^{2} \)
89 \( 1 + 3.95T + 7.92e3T^{2} \)
97 \( 1 + (10.2 - 31.5i)T + (-7.61e3 - 5.53e3i)T^{2} \)
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

−16.21915636426053968283197854243, −14.81730466521156678434884551797, −14.39027028765286439415730909263, −12.93816406211468094456494304268, −12.09802729369145166523535338197, −9.893197842769692587878035916247, −8.921279076910632092955032935317, −6.82738574982990943148784938446, −5.49834891209103864267408730009, −3.21440352003552587764182549672, 3.49922187538414522079252699550, 4.82791462362337011256139196656, 7.16200044991718095372141469654, 8.975238162432073582552533504063, 9.996028562044816719804292107813, 11.91950352323406515877155211904, 13.07191995243509538863667006577, 13.93786686138610365865207270573, 14.97866677237153846474842673887, 16.78917834248055619758469324584

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