Properties

Label 2-33-33.17-c5-0-11
Degree $2$
Conductor $33$
Sign $0.734 + 0.678i$
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  + (−0.489 − 1.50i)2-s + (15.5 − 0.0191i)3-s + (23.8 − 17.3i)4-s + (−16.6 − 5.40i)5-s + (−7.66 − 23.4i)6-s + (36.5 + 50.2i)7-s + (−78.8 − 57.3i)8-s + (242. − 0.598i)9-s + 27.7i·10-s + (56.2 − 397. i)11-s + (371. − 270. i)12-s + (283. − 92.1i)13-s + (57.9 − 79.7i)14-s + (−259. − 83.9i)15-s + (243. − 750. i)16-s + (−320. + 985. i)17-s + ⋯
L(s)  = 1  + (−0.0866 − 0.266i)2-s + (0.999 − 0.00123i)3-s + (0.745 − 0.541i)4-s + (−0.297 − 0.0967i)5-s + (−0.0869 − 0.266i)6-s + (0.281 + 0.387i)7-s + (−0.435 − 0.316i)8-s + (0.999 − 0.00246i)9-s + 0.0877i·10-s + (0.140 − 0.990i)11-s + (0.744 − 0.542i)12-s + (0.465 − 0.151i)13-s + (0.0790 − 0.108i)14-s + (−0.297 − 0.0963i)15-s + (0.238 − 0.732i)16-s + (−0.268 + 0.826i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.734 + 0.678i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.734 + 0.678i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $0.734 + 0.678i$
Analytic conductor: \(5.29266\)
Root analytic conductor: \(2.30057\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{33} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 33,\ (\ :5/2),\ 0.734 + 0.678i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.08208 - 0.814495i\)
\(L(\frac12)\) \(\approx\) \(2.08208 - 0.814495i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-15.5 + 0.0191i)T \)
11 \( 1 + (-56.2 + 397. i)T \)
good2 \( 1 + (0.489 + 1.50i)T + (-25.8 + 18.8i)T^{2} \)
5 \( 1 + (16.6 + 5.40i)T + (2.52e3 + 1.83e3i)T^{2} \)
7 \( 1 + (-36.5 - 50.2i)T + (-5.19e3 + 1.59e4i)T^{2} \)
13 \( 1 + (-283. + 92.1i)T + (3.00e5 - 2.18e5i)T^{2} \)
17 \( 1 + (320. - 985. i)T + (-1.14e6 - 8.34e5i)T^{2} \)
19 \( 1 + (821. - 1.13e3i)T + (-7.65e5 - 2.35e6i)T^{2} \)
23 \( 1 - 3.74e3iT - 6.43e6T^{2} \)
29 \( 1 + (5.35e3 - 3.88e3i)T + (6.33e6 - 1.95e7i)T^{2} \)
31 \( 1 + (153. + 473. i)T + (-2.31e7 + 1.68e7i)T^{2} \)
37 \( 1 + (-219. + 159. i)T + (2.14e7 - 6.59e7i)T^{2} \)
41 \( 1 + (-3.71e3 - 2.69e3i)T + (3.58e7 + 1.10e8i)T^{2} \)
43 \( 1 - 1.41e4iT - 1.47e8T^{2} \)
47 \( 1 + (-2.15e3 + 2.96e3i)T + (-7.08e7 - 2.18e8i)T^{2} \)
53 \( 1 + (-1.25e4 + 4.07e3i)T + (3.38e8 - 2.45e8i)T^{2} \)
59 \( 1 + (-5.22e3 - 7.19e3i)T + (-2.20e8 + 6.79e8i)T^{2} \)
61 \( 1 + (2.29e4 + 7.46e3i)T + (6.83e8 + 4.96e8i)T^{2} \)
67 \( 1 + 5.39e4T + 1.35e9T^{2} \)
71 \( 1 + (9.42e3 + 3.06e3i)T + (1.45e9 + 1.06e9i)T^{2} \)
73 \( 1 + (-3.52e3 - 4.85e3i)T + (-6.40e8 + 1.97e9i)T^{2} \)
79 \( 1 + (-9.39e4 + 3.05e4i)T + (2.48e9 - 1.80e9i)T^{2} \)
83 \( 1 + (-3.57e4 + 1.09e5i)T + (-3.18e9 - 2.31e9i)T^{2} \)
89 \( 1 + 1.13e5iT - 5.58e9T^{2} \)
97 \( 1 + (-5.21e3 - 1.60e4i)T + (-6.94e9 + 5.04e9i)T^{2} \)
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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

−15.37990184561792851274585549787, −14.54768618215133454454555281789, −13.19114309537160404193211876510, −11.70845535997115909153644143684, −10.50845932876934599499600641559, −9.044483128247583659442611043222, −7.79091188385451032453404192387, −5.99643419461931580993043732793, −3.52096524830232460608359409113, −1.70168378988997049458107773359, 2.31030943900096756436553060250, 4.11137718623348216073468893766, 6.87794928474803836555944426203, 7.79307942950966918112328621825, 9.135820918646697187442861348886, 10.82666693175058567839136361363, 12.22937607259967623251610737654, 13.51422139526853041710861928387, 14.90058697661824797520463899951, 15.60490038821555525941579756537

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