Properties

Degree $2$
Conductor $33$
Sign $0.503 + 0.863i$
Motivic weight $4$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.57i·2-s − 5.19·3-s + 13.5·4-s + 15.5·5-s + 8.18i·6-s − 93.8i·7-s − 46.4i·8-s + 27·9-s − 24.4i·10-s + (60.9 + 104. i)11-s − 70.2·12-s + 29.4i·13-s − 147.·14-s − 80.8·15-s + 143.·16-s + 251. i·17-s + ⋯
L(s)  = 1  − 0.393i·2-s − 0.577·3-s + 0.845·4-s + 0.622·5-s + 0.227i·6-s − 1.91i·7-s − 0.726i·8-s + 0.333·9-s − 0.244i·10-s + (0.503 + 0.863i)11-s − 0.487·12-s + 0.174i·13-s − 0.753·14-s − 0.359·15-s + 0.559·16-s + 0.871i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.503 + 0.863i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.503 + 0.863i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $0.503 + 0.863i$
Motivic weight: \(4\)
Character: $\chi_{33} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 33,\ (\ :2),\ 0.503 + 0.863i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.33134 - 0.764949i\)
\(L(\frac12)\) \(\approx\) \(1.33134 - 0.764949i\)
\(L(3)\) 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 \( 1 + (-60.9 - 104. i)T \)
good2 \( 1 + 1.57iT - 16T^{2} \)
5 \( 1 - 15.5T + 625T^{2} \)
7 \( 1 + 93.8iT - 2.40e3T^{2} \)
13 \( 1 - 29.4iT - 2.85e4T^{2} \)
17 \( 1 - 251. iT - 8.35e4T^{2} \)
19 \( 1 + 80.2iT - 1.30e5T^{2} \)
23 \( 1 + 702.T + 2.79e5T^{2} \)
29 \( 1 - 1.44e3iT - 7.07e5T^{2} \)
31 \( 1 - 1.27e3T + 9.23e5T^{2} \)
37 \( 1 - 115.T + 1.87e6T^{2} \)
41 \( 1 + 1.07e3iT - 2.82e6T^{2} \)
43 \( 1 - 1.88e3iT - 3.41e6T^{2} \)
47 \( 1 - 1.59e3T + 4.87e6T^{2} \)
53 \( 1 - 1.19e3T + 7.89e6T^{2} \)
59 \( 1 - 1.89e3T + 1.21e7T^{2} \)
61 \( 1 - 3.77e3iT - 1.38e7T^{2} \)
67 \( 1 + 1.25e3T + 2.01e7T^{2} \)
71 \( 1 + 3.59e3T + 2.54e7T^{2} \)
73 \( 1 - 1.75e3iT - 2.83e7T^{2} \)
79 \( 1 + 6.74e3iT - 3.89e7T^{2} \)
83 \( 1 + 8.61e3iT - 4.74e7T^{2} \)
89 \( 1 + 9.33e3T + 6.27e7T^{2} \)
97 \( 1 - 1.10e4T + 8.85e7T^{2} \)
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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

−16.04599248891707339196775559843, −14.44502453138229016805963648739, −13.18771131278967308429831468468, −11.92528399725588767282921213463, −10.58354136467132796912096990014, −10.01050514036815067638076561226, −7.39972936833355495526211769984, −6.37336688074956506413783619405, −4.09881976656108944280268641503, −1.45791979145342440924391605888, 2.36244205006677585407388664530, 5.64579547093016140168069476546, 6.23236901964968190106005010021, 8.251812965856337433546770268030, 9.775814425412775151704572684398, 11.54403940748133327485714022329, 12.07230732849755557013644084127, 13.89787185906217091731153018353, 15.33587206931222080139910234112, 16.00172801104246892442716132312

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