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Properties

Label 2-33-3.2-c4-0-0
Degree $2$
Conductor $33$
Sign $-0.409 + 0.912i$
Analytic cond. $3.41120$
Root an. cond. $1.84694$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 6.53i·2-s + (−8.21 − 3.68i)3-s − 26.6·4-s − 12.2i·5-s + (24.0 − 53.6i)6-s − 55.6·7-s − 69.8i·8-s + (53.8 + 60.4i)9-s + 79.9·10-s − 36.4i·11-s + (219. + 98.2i)12-s − 240.·13-s − 363. i·14-s + (−45.0 + 100. i)15-s + 29.5·16-s + 364. i·17-s + ⋯
L(s)  = 1  + 1.63i·2-s + (−0.912 − 0.409i)3-s − 1.66·4-s − 0.489i·5-s + (0.668 − 1.49i)6-s − 1.13·7-s − 1.09i·8-s + (0.665 + 0.746i)9-s + 0.799·10-s − 0.301i·11-s + (1.52 + 0.682i)12-s − 1.42·13-s − 1.85i·14-s + (−0.200 + 0.446i)15-s + 0.115·16-s + 1.26i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.409 + 0.912i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.409 + 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $-0.409 + 0.912i$
Analytic conductor: \(3.41120\)
Root analytic conductor: \(1.84694\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{33} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 33,\ (\ :2),\ -0.409 + 0.912i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.0954695 - 0.147427i\)
\(L(\frac12)\) \(\approx\) \(0.0954695 - 0.147427i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (8.21 + 3.68i)T \)
11 \( 1 + 36.4iT \)
good2 \( 1 - 6.53iT - 16T^{2} \)
5 \( 1 + 12.2iT - 625T^{2} \)
7 \( 1 + 55.6T + 2.40e3T^{2} \)
13 \( 1 + 240.T + 2.85e4T^{2} \)
17 \( 1 - 364. iT - 8.35e4T^{2} \)
19 \( 1 + 230.T + 1.30e5T^{2} \)
23 \( 1 + 193. iT - 2.79e5T^{2} \)
29 \( 1 - 1.40e3iT - 7.07e5T^{2} \)
31 \( 1 - 1.05e3T + 9.23e5T^{2} \)
37 \( 1 - 166.T + 1.87e6T^{2} \)
41 \( 1 + 2.13e3iT - 2.82e6T^{2} \)
43 \( 1 + 3.38e3T + 3.41e6T^{2} \)
47 \( 1 + 98.7iT - 4.87e6T^{2} \)
53 \( 1 - 2.13e3iT - 7.89e6T^{2} \)
59 \( 1 + 646. iT - 1.21e7T^{2} \)
61 \( 1 - 646.T + 1.38e7T^{2} \)
67 \( 1 + 6.26e3T + 2.01e7T^{2} \)
71 \( 1 - 2.44e3iT - 2.54e7T^{2} \)
73 \( 1 + 565.T + 2.83e7T^{2} \)
79 \( 1 + 5.75e3T + 3.89e7T^{2} \)
83 \( 1 + 4.76e3iT - 4.74e7T^{2} \)
89 \( 1 + 1.14e4iT - 6.27e7T^{2} \)
97 \( 1 + 3.56e3T + 8.85e7T^{2} \)
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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

−16.80326966480671776000196090016, −15.91697558890196966281208942595, −14.69212214390468886368719728746, −13.16433156819274541363114928107, −12.36310023617941681957672529337, −10.26906858154488119281568583776, −8.643505980642963047143504948553, −7.11632991776104376037975803832, −6.17512742883273223798129431305, −4.85517156590110675675082396981, 0.13576013610252794419969532596, 2.88375208383835702445690962722, 4.63791400446325096045324781152, 6.74245216477623617971947658490, 9.673651315757509721421756445541, 10.03872388965037832793811938239, 11.45302476701784715935928000329, 12.25639818442348890419387877490, 13.34581026814574335831993708948, 15.13172481029193432667800907600

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