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Properties

Label 2-33-11.10-c6-0-6
Degree $2$
Conductor $33$
Sign $0.917 - 0.397i$
Analytic cond. $7.59178$
Root an. cond. $2.75531$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.61i·2-s + 15.5·3-s + 50.9·4-s + 42.6·5-s + 56.2i·6-s − 392. i·7-s + 415. i·8-s + 243·9-s + 154. i·10-s + (1.22e3 − 529. i)11-s + 794.·12-s + 2.61e3i·13-s + 1.41e3·14-s + 665.·15-s + 1.76e3·16-s + 7.83e3i·17-s + ⋯
L(s)  = 1  + 0.451i·2-s + 0.577·3-s + 0.796·4-s + 0.341·5-s + 0.260i·6-s − 1.14i·7-s + 0.810i·8-s + 0.333·9-s + 0.154i·10-s + (0.917 − 0.397i)11-s + 0.459·12-s + 1.18i·13-s + 0.516·14-s + 0.197·15-s + 0.430·16-s + 1.59i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.917 - 0.397i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.917 - 0.397i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $0.917 - 0.397i$
Analytic conductor: \(7.59178\)
Root analytic conductor: \(2.75531\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{33} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 33,\ (\ :3),\ 0.917 - 0.397i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(2.46615 + 0.511689i\)
\(L(\frac12)\) \(\approx\) \(2.46615 + 0.511689i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 15.5T \)
11 \( 1 + (-1.22e3 + 529. i)T \)
good2 \( 1 - 3.61iT - 64T^{2} \)
5 \( 1 - 42.6T + 1.56e4T^{2} \)
7 \( 1 + 392. iT - 1.17e5T^{2} \)
13 \( 1 - 2.61e3iT - 4.82e6T^{2} \)
17 \( 1 - 7.83e3iT - 2.41e7T^{2} \)
19 \( 1 + 9.64e3iT - 4.70e7T^{2} \)
23 \( 1 + 1.96e3T + 1.48e8T^{2} \)
29 \( 1 + 2.88e4iT - 5.94e8T^{2} \)
31 \( 1 + 3.11e4T + 8.87e8T^{2} \)
37 \( 1 + 7.13e4T + 2.56e9T^{2} \)
41 \( 1 - 7.18e4iT - 4.75e9T^{2} \)
43 \( 1 + 1.03e5iT - 6.32e9T^{2} \)
47 \( 1 - 5.34e4T + 1.07e10T^{2} \)
53 \( 1 - 5.78e4T + 2.21e10T^{2} \)
59 \( 1 + 1.28e5T + 4.21e10T^{2} \)
61 \( 1 - 5.90e4iT - 5.15e10T^{2} \)
67 \( 1 + 2.08e5T + 9.04e10T^{2} \)
71 \( 1 + 6.21e5T + 1.28e11T^{2} \)
73 \( 1 + 1.25e5iT - 1.51e11T^{2} \)
79 \( 1 - 4.67e4iT - 2.43e11T^{2} \)
83 \( 1 - 4.96e5iT - 3.26e11T^{2} \)
89 \( 1 + 1.66e5T + 4.96e11T^{2} \)
97 \( 1 - 1.49e6T + 8.32e11T^{2} \)
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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

−15.45375337134383867379965987773, −14.33070821055581571378099056462, −13.47384023015481220187348264422, −11.67950438710992875384534334381, −10.46436290327495887316449545751, −8.886670762057183771284047383554, −7.36051569250276911630130841683, −6.31117442659694773109810928014, −3.94192123976492620428799955276, −1.77784599995074291590159499694, 1.80968805233278152121600546465, 3.22050055387751544524956502636, 5.73545363345003556240599716424, 7.38930539604197212911733987884, 9.067515931107252248993669306770, 10.22499024806828519095934271779, 11.81961218068049589512877987394, 12.60561683078507801553386577309, 14.26450158996688714235121170020, 15.31718994766065509387057522591

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