Properties

Degree $2$
Conductor $33$
Sign $-0.989 - 0.143i$
Motivic weight $4$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 7.70i·2-s + 5.19·3-s − 43.3·4-s − 12.5·5-s − 40.0i·6-s − 63.6i·7-s + 210. i·8-s + 27·9-s + 96.4i·10-s + (119. + 17.3i)11-s − 225.·12-s − 194. i·13-s − 490.·14-s − 65.1·15-s + 926.·16-s − 108. i·17-s + ⋯
L(s)  = 1  − 1.92i·2-s + 0.577·3-s − 2.70·4-s − 0.501·5-s − 1.11i·6-s − 1.29i·7-s + 3.28i·8-s + 0.333·9-s + 0.964i·10-s + (0.989 + 0.143i)11-s − 1.56·12-s − 1.15i·13-s − 2.50·14-s − 0.289·15-s + 3.61·16-s − 0.374i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 - 0.143i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.989 - 0.143i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $-0.989 - 0.143i$
Motivic weight: \(4\)
Character: $\chi_{33} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 33,\ (\ :2),\ -0.989 - 0.143i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.0909923 + 1.26027i\)
\(L(\frac12)\) \(\approx\) \(0.0909923 + 1.26027i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 5.19T \)
11 \( 1 + (-119. - 17.3i)T \)
good2 \( 1 + 7.70iT - 16T^{2} \)
5 \( 1 + 12.5T + 625T^{2} \)
7 \( 1 + 63.6iT - 2.40e3T^{2} \)
13 \( 1 + 194. iT - 2.85e4T^{2} \)
17 \( 1 + 108. iT - 8.35e4T^{2} \)
19 \( 1 - 69.0iT - 1.30e5T^{2} \)
23 \( 1 - 576.T + 2.79e5T^{2} \)
29 \( 1 - 382. iT - 7.07e5T^{2} \)
31 \( 1 + 36.6T + 9.23e5T^{2} \)
37 \( 1 - 1.79e3T + 1.87e6T^{2} \)
41 \( 1 - 2.88e3iT - 2.82e6T^{2} \)
43 \( 1 - 319. iT - 3.41e6T^{2} \)
47 \( 1 - 2.29e3T + 4.87e6T^{2} \)
53 \( 1 - 857.T + 7.89e6T^{2} \)
59 \( 1 + 2.14e3T + 1.21e7T^{2} \)
61 \( 1 + 4.96e3iT - 1.38e7T^{2} \)
67 \( 1 + 5.36e3T + 2.01e7T^{2} \)
71 \( 1 + 4.95e3T + 2.54e7T^{2} \)
73 \( 1 + 3.58e3iT - 2.83e7T^{2} \)
79 \( 1 - 7.14e3iT - 3.89e7T^{2} \)
83 \( 1 + 156. iT - 4.74e7T^{2} \)
89 \( 1 - 7.18e3T + 6.27e7T^{2} \)
97 \( 1 - 2.41e3T + 8.85e7T^{2} \)
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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

−14.78320978619918907192048459712, −13.68340966600695927774643915266, −12.77805278703113820431271303501, −11.46987222002136079993600069815, −10.42231303248119568561531928371, −9.319756019496587586228768030933, −7.81627522686754122638449434737, −4.39227213821317594808471829869, −3.24538360669046602277161269502, −0.991515655656918157829289301245, 4.20468863388086682595556654100, 5.95999018942273204016869174887, 7.27442111591566808003601560608, 8.710557429072235770346262648365, 9.247700675708319796598308528157, 12.04837661136944494080249945859, 13.54428125129357737702887211354, 14.67369354664865299760270942820, 15.30322859855852239891367342704, 16.29823200965018039386893654888

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