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Properties

Label 2-33-33.32-c7-0-21
Degree $2$
Conductor $33$
Sign $-0.132 + 0.991i$
Analytic cond. $10.3087$
Root an. cond. $3.21071$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 17.1·2-s + (−43.8 − 16.3i)3-s + 167.·4-s − 90.8i·5-s + (−753. − 281. i)6-s − 1.50e3i·7-s + 678.·8-s + (1.65e3 + 1.43e3i)9-s − 1.56e3i·10-s + (−981. − 4.30e3i)11-s + (−7.33e3 − 2.73e3i)12-s − 3.56e3i·13-s − 2.58e4i·14-s + (−1.48e3 + 3.98e3i)15-s − 9.77e3·16-s + 1.34e4·17-s + ⋯
L(s)  = 1  + 1.51·2-s + (−0.936 − 0.349i)3-s + 1.30·4-s − 0.325i·5-s + (−1.42 − 0.531i)6-s − 1.65i·7-s + 0.468·8-s + (0.755 + 0.655i)9-s − 0.494i·10-s + (−0.222 − 0.974i)11-s + (−1.22 − 0.457i)12-s − 0.450i·13-s − 2.52i·14-s + (−0.113 + 0.304i)15-s − 0.596·16-s + 0.664·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.132 + 0.991i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.132 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $-0.132 + 0.991i$
Analytic conductor: \(10.3087\)
Root analytic conductor: \(3.21071\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{33} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 33,\ (\ :7/2),\ -0.132 + 0.991i)\)

Particular Values

\(L(4)\) \(\approx\) \(1.69684 - 1.93928i\)
\(L(\frac12)\) \(\approx\) \(1.69684 - 1.93928i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (43.8 + 16.3i)T \)
11 \( 1 + (981. + 4.30e3i)T \)
good2 \( 1 - 17.1T + 128T^{2} \)
5 \( 1 + 90.8iT - 7.81e4T^{2} \)
7 \( 1 + 1.50e3iT - 8.23e5T^{2} \)
13 \( 1 + 3.56e3iT - 6.27e7T^{2} \)
17 \( 1 - 1.34e4T + 4.10e8T^{2} \)
19 \( 1 - 2.67e4iT - 8.93e8T^{2} \)
23 \( 1 - 4.04e4iT - 3.40e9T^{2} \)
29 \( 1 - 1.42e5T + 1.72e10T^{2} \)
31 \( 1 + 1.58e5T + 2.75e10T^{2} \)
37 \( 1 - 2.86e5T + 9.49e10T^{2} \)
41 \( 1 - 4.34e5T + 1.94e11T^{2} \)
43 \( 1 - 6.89e5iT - 2.71e11T^{2} \)
47 \( 1 + 1.28e6iT - 5.06e11T^{2} \)
53 \( 1 + 4.06e5iT - 1.17e12T^{2} \)
59 \( 1 + 6.22e5iT - 2.48e12T^{2} \)
61 \( 1 - 9.01e5iT - 3.14e12T^{2} \)
67 \( 1 - 2.03e6T + 6.06e12T^{2} \)
71 \( 1 + 2.52e6iT - 9.09e12T^{2} \)
73 \( 1 + 1.13e6iT - 1.10e13T^{2} \)
79 \( 1 + 1.22e6iT - 1.92e13T^{2} \)
83 \( 1 - 5.90e6T + 2.71e13T^{2} \)
89 \( 1 + 7.09e6iT - 4.42e13T^{2} \)
97 \( 1 - 8.09e6T + 8.07e13T^{2} \)
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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

−14.41807683993877751374637553248, −13.44755382009846361014864313850, −12.70396944622856869979947310161, −11.41338289397005113063854787917, −10.38106071666736431527521056573, −7.63231733974944600826388961589, −6.23253151902780713357914584244, −5.01228594006963307390708164501, −3.64502723914876290972954803597, −0.859229719073111302890218593868, 2.61241901686184578486604402332, 4.56602917676942999322115260917, 5.58870120898019959028756404199, 6.73672383363712939531083618273, 9.306580834149487908589761996281, 11.04724681217179118660985002102, 12.19386979778707440001196023561, 12.66906369694345031946317892477, 14.51498659048778283595805288375, 15.28276819826158572303476209819

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