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Properties

Label 2-33-33.32-c7-0-0
Degree $2$
Conductor $33$
Sign $-0.549 - 0.835i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.549 - 0.835i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.549 - 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $-0.549 - 0.835i$
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: \((2,\ 33,\ (\ :7/2),\ -0.549 - 0.835i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.107717 + 0.199746i\)
\(L(\frac12)\) \(\approx\) \(0.107717 + 0.199746i\)
\(L(\frac{9}{2})\) 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

−16.26748282007290077144848643404, −14.73811104095093266339437669167, −12.88967348368739542566793833921, −11.12497837401908299619781554560, −10.45767005408752663932951711045, −9.562584329953384933478996530217, −7.62194695128327231118815443875, −6.69889527203250000735283780292, −4.32248978498637462327321548245, −1.20273375360849235274539692208, 0.22447815832029443851583594557, 1.93088967290054575796467360221, 5.39897079696896344244423101584, 6.83677742660111769836635021457, 8.519616562835715344582212814111, 9.330229346804296199071517283099, 11.03991285360539487675146493915, 11.75778268257753860567774170003, 13.20068243855399506097695744944, 15.40640947747665421771182335692

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