Properties

Label 2-33-33.2-c5-0-3
Degree $2$
Conductor $33$
Sign $-0.790 + 0.612i$
Analytic cond. $5.29266$
Root an. cond. $2.30057$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.81 + 8.67i)2-s + (12.9 + 8.66i)3-s + (−41.4 − 30.1i)4-s + (−53.2 + 17.2i)5-s + (−111. + 87.9i)6-s + (−14.1 + 19.4i)7-s + (142. − 103. i)8-s + (92.7 + 224. i)9-s − 510. i·10-s + (−240. − 321. i)11-s + (−276. − 749. i)12-s + (323. + 105. i)13-s + (−128. − 177. i)14-s + (−839. − 237. i)15-s + (−11.3 − 34.9i)16-s + (463. + 1.42e3i)17-s + ⋯
L(s)  = 1  + (−0.498 + 1.53i)2-s + (0.831 + 0.556i)3-s + (−1.29 − 0.941i)4-s + (−0.952 + 0.309i)5-s + (−1.26 + 0.997i)6-s + (−0.108 + 0.149i)7-s + (0.785 − 0.570i)8-s + (0.381 + 0.924i)9-s − 1.61i·10-s + (−0.599 − 0.800i)11-s + (−0.553 − 1.50i)12-s + (0.531 + 0.172i)13-s + (−0.175 − 0.241i)14-s + (−0.963 − 0.272i)15-s + (−0.0111 − 0.0341i)16-s + (0.389 + 1.19i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.790 + 0.612i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.790 + 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $-0.790 + 0.612i$
Analytic conductor: \(5.29266\)
Root analytic conductor: \(2.30057\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{33} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 33,\ (\ :5/2),\ -0.790 + 0.612i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.309347 - 0.903756i\)
\(L(\frac12)\) \(\approx\) \(0.309347 - 0.903756i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-12.9 - 8.66i)T \)
11 \( 1 + (240. + 321. i)T \)
good2 \( 1 + (2.81 - 8.67i)T + (-25.8 - 18.8i)T^{2} \)
5 \( 1 + (53.2 - 17.2i)T + (2.52e3 - 1.83e3i)T^{2} \)
7 \( 1 + (14.1 - 19.4i)T + (-5.19e3 - 1.59e4i)T^{2} \)
13 \( 1 + (-323. - 105. i)T + (3.00e5 + 2.18e5i)T^{2} \)
17 \( 1 + (-463. - 1.42e3i)T + (-1.14e6 + 8.34e5i)T^{2} \)
19 \( 1 + (-167. - 230. i)T + (-7.65e5 + 2.35e6i)T^{2} \)
23 \( 1 - 4.34e3iT - 6.43e6T^{2} \)
29 \( 1 + (6.12e3 + 4.45e3i)T + (6.33e6 + 1.95e7i)T^{2} \)
31 \( 1 + (937. - 2.88e3i)T + (-2.31e7 - 1.68e7i)T^{2} \)
37 \( 1 + (-1.01e4 - 7.37e3i)T + (2.14e7 + 6.59e7i)T^{2} \)
41 \( 1 + (-7.65e3 + 5.56e3i)T + (3.58e7 - 1.10e8i)T^{2} \)
43 \( 1 + 8.56e3iT - 1.47e8T^{2} \)
47 \( 1 + (-7.90e3 - 1.08e4i)T + (-7.08e7 + 2.18e8i)T^{2} \)
53 \( 1 + (1.30e4 + 4.23e3i)T + (3.38e8 + 2.45e8i)T^{2} \)
59 \( 1 + (-2.41e4 + 3.33e4i)T + (-2.20e8 - 6.79e8i)T^{2} \)
61 \( 1 + (-4.35e4 + 1.41e4i)T + (6.83e8 - 4.96e8i)T^{2} \)
67 \( 1 - 2.23e3T + 1.35e9T^{2} \)
71 \( 1 + (8.80e3 - 2.86e3i)T + (1.45e9 - 1.06e9i)T^{2} \)
73 \( 1 + (-2.09e4 + 2.88e4i)T + (-6.40e8 - 1.97e9i)T^{2} \)
79 \( 1 + (3.50e4 + 1.13e4i)T + (2.48e9 + 1.80e9i)T^{2} \)
83 \( 1 + (-2.00e3 - 6.18e3i)T + (-3.18e9 + 2.31e9i)T^{2} \)
89 \( 1 + 4.56e3iT - 5.58e9T^{2} \)
97 \( 1 + (2.55e4 - 7.87e4i)T + (-6.94e9 - 5.04e9i)T^{2} \)
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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

−15.96913832372506719869010452192, −15.51562841403611698061381561777, −14.57466886226354080121071154711, −13.37722654082842722825000234935, −11.15033514777701344944566955643, −9.528439196769950255107174006229, −8.250188121342875993572904131079, −7.59523441273515226511229158775, −5.72390541923860697207059470639, −3.68914891142494102394758527921, 0.63145543464729386660566854480, 2.54879842149092068100124403975, 4.05284686825767719838755221651, 7.44475874653741423137158295422, 8.653625742808554877063516010873, 9.839464974600678051787825232994, 11.30616415800473285936761825573, 12.43955408094776801303418645219, 13.14461826965265003366438124299, 14.77557227175690708262158866216

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