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Properties

Label 2-33-3.2-c8-0-10
Degree $2$
Conductor $33$
Sign $-0.972 + 0.231i$
Analytic cond. $13.4434$
Root an. cond. $3.66653$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 17.8i·2-s + (18.7 + 78.8i)3-s − 61.9·4-s + 794. i·5-s + (−1.40e3 + 334. i)6-s + 2.43e3·7-s + 3.45e3i·8-s + (−5.85e3 + 2.95e3i)9-s − 1.41e4·10-s + 4.41e3i·11-s + (−1.16e3 − 4.88e3i)12-s + 4.37e4·13-s + 4.34e4i·14-s + (−6.26e4 + 1.48e4i)15-s − 7.75e4·16-s − 1.49e5i·17-s + ⋯
L(s)  = 1  + 1.11i·2-s + (0.231 + 0.972i)3-s − 0.242·4-s + 1.27i·5-s + (−1.08 + 0.257i)6-s + 1.01·7-s + 0.844i·8-s + (−0.893 + 0.449i)9-s − 1.41·10-s + 0.301i·11-s + (−0.0559 − 0.235i)12-s + 1.53·13-s + 1.13i·14-s + (−1.23 + 0.293i)15-s − 1.18·16-s − 1.79i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 + 0.231i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.972 + 0.231i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $-0.972 + 0.231i$
Analytic conductor: \(13.4434\)
Root analytic conductor: \(3.66653\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{33} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 33,\ (\ :4),\ -0.972 + 0.231i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.267769 - 2.28446i\)
\(L(\frac12)\) \(\approx\) \(0.267769 - 2.28446i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-18.7 - 78.8i)T \)
11 \( 1 - 4.41e3iT \)
good2 \( 1 - 17.8iT - 256T^{2} \)
5 \( 1 - 794. iT - 3.90e5T^{2} \)
7 \( 1 - 2.43e3T + 5.76e6T^{2} \)
13 \( 1 - 4.37e4T + 8.15e8T^{2} \)
17 \( 1 + 1.49e5iT - 6.97e9T^{2} \)
19 \( 1 + 1.74e4T + 1.69e10T^{2} \)
23 \( 1 + 4.63e5iT - 7.83e10T^{2} \)
29 \( 1 - 4.99e5iT - 5.00e11T^{2} \)
31 \( 1 - 1.37e5T + 8.52e11T^{2} \)
37 \( 1 - 5.64e5T + 3.51e12T^{2} \)
41 \( 1 + 3.76e5iT - 7.98e12T^{2} \)
43 \( 1 - 5.67e6T + 1.16e13T^{2} \)
47 \( 1 - 9.56e6iT - 2.38e13T^{2} \)
53 \( 1 + 4.43e6iT - 6.22e13T^{2} \)
59 \( 1 + 2.31e6iT - 1.46e14T^{2} \)
61 \( 1 + 2.29e7T + 1.91e14T^{2} \)
67 \( 1 - 7.17e6T + 4.06e14T^{2} \)
71 \( 1 - 3.78e6iT - 6.45e14T^{2} \)
73 \( 1 + 9.80e6T + 8.06e14T^{2} \)
79 \( 1 - 1.13e6T + 1.51e15T^{2} \)
83 \( 1 + 2.95e7iT - 2.25e15T^{2} \)
89 \( 1 + 5.92e7iT - 3.93e15T^{2} \)
97 \( 1 - 7.89e7T + 7.83e15T^{2} \)
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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

−15.57582469813624888767751600726, −14.44679875184770275183613750452, −14.13618916298088666562317268816, −11.32063334089436037209164169286, −10.73176950512624799364520146670, −8.876394145881647005532139055439, −7.61467325195136549456370439151, −6.23754130988543656672087118944, −4.70063842732658880954119541833, −2.72483705569882979905979496432, 1.06113454910238854219430288537, 1.72445895134968769410061466573, 3.84228340758816966863074175690, 5.93042654138818907570867628387, 8.002049013628538307475870692737, 8.947773934151861401803540756043, 10.91372619410166541631814624937, 11.84699962335441185136319990659, 12.89651926570944800168976073100, 13.63685339250957481447173517625

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