Properties

Degree $2$
Conductor $33$
Sign $-0.830 + 0.557i$
Motivic weight $4$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.58i·2-s − 5.19·3-s − 15.1·4-s − 29.7·5-s − 29.0i·6-s − 12.9i·7-s + 4.47i·8-s + 27·9-s − 166. i·10-s + (−100. + 67.4i)11-s + 78.9·12-s − 36.6i·13-s + 72.2·14-s + 154.·15-s − 268.·16-s + 464. i·17-s + ⋯
L(s)  = 1  + 1.39i·2-s − 0.577·3-s − 0.949·4-s − 1.18·5-s − 0.806i·6-s − 0.264i·7-s + 0.0698i·8-s + 0.333·9-s − 1.66i·10-s + (−0.830 + 0.557i)11-s + 0.548·12-s − 0.216i·13-s + 0.368·14-s + 0.687·15-s − 1.04·16-s + 1.60i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.830 + 0.557i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.830 + 0.557i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $-0.830 + 0.557i$
Motivic weight: \(4\)
Character: $\chi_{33} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 33,\ (\ :2),\ -0.830 + 0.557i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.157914 - 0.518171i\)
\(L(\frac12)\) \(\approx\) \(0.157914 - 0.518171i\)
\(L(3)\) 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 \( 1 + (100. - 67.4i)T \)
good2 \( 1 - 5.58iT - 16T^{2} \)
5 \( 1 + 29.7T + 625T^{2} \)
7 \( 1 + 12.9iT - 2.40e3T^{2} \)
13 \( 1 + 36.6iT - 2.85e4T^{2} \)
17 \( 1 - 464. iT - 8.35e4T^{2} \)
19 \( 1 - 327. iT - 1.30e5T^{2} \)
23 \( 1 - 396.T + 2.79e5T^{2} \)
29 \( 1 + 1.15e3iT - 7.07e5T^{2} \)
31 \( 1 - 437.T + 9.23e5T^{2} \)
37 \( 1 - 276.T + 1.87e6T^{2} \)
41 \( 1 - 2.78e3iT - 2.82e6T^{2} \)
43 \( 1 + 1.66e3iT - 3.41e6T^{2} \)
47 \( 1 + 1.85e3T + 4.87e6T^{2} \)
53 \( 1 - 3.74e3T + 7.89e6T^{2} \)
59 \( 1 + 6.57e3T + 1.21e7T^{2} \)
61 \( 1 - 5.14e3iT - 1.38e7T^{2} \)
67 \( 1 + 3.46e3T + 2.01e7T^{2} \)
71 \( 1 + 6.03e3T + 2.54e7T^{2} \)
73 \( 1 - 3.58e3iT - 2.83e7T^{2} \)
79 \( 1 - 9.71e3iT - 3.89e7T^{2} \)
83 \( 1 + 1.18e4iT - 4.74e7T^{2} \)
89 \( 1 - 2.58e3T + 6.27e7T^{2} \)
97 \( 1 - 1.55e3T + 8.85e7T^{2} \)
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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

−16.54740223139641928593093745542, −15.45040218148692674548079563583, −14.94399648930167652512609377375, −13.16735446576011077642377366865, −11.81217244566027900373039877889, −10.42007192871083289145996850636, −8.238345459531409391782889250902, −7.43619867260999712851446398765, −5.96618750392371359288711092566, −4.37054655346817050757255810574, 0.41487617480888194126538289288, 3.04645909623046377122854596137, 4.81129686017761205338950136874, 7.22522899829977447971499077824, 9.077544503380697826856464630967, 10.71211818117809503166568304418, 11.47087644907903782264864217241, 12.29965043916977721682914821166, 13.52910293503173915948083375567, 15.50715904263705374030557485149

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