Properties

Label 2-33-3.2-c4-0-9
Degree $2$
Conductor $33$
Sign $0.496 + 0.868i$
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  − 2.36i·2-s + (7.81 − 4.46i)3-s + 10.4·4-s + 9.92i·5-s + (−10.5 − 18.4i)6-s − 13.3·7-s − 62.3i·8-s + (41.1 − 69.7i)9-s + 23.4·10-s + 36.4i·11-s + (81.4 − 46.5i)12-s − 138.·13-s + 31.4i·14-s + (44.3 + 77.5i)15-s + 19.5·16-s + 445. i·17-s + ⋯
L(s)  = 1  − 0.590i·2-s + (0.868 − 0.496i)3-s + 0.651·4-s + 0.397i·5-s + (−0.292 − 0.512i)6-s − 0.272·7-s − 0.974i·8-s + (0.507 − 0.861i)9-s + 0.234·10-s + 0.301i·11-s + (0.565 − 0.323i)12-s − 0.820·13-s + 0.160i·14-s + (0.197 + 0.344i)15-s + 0.0765·16-s + 1.54i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.496 + 0.868i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.496 + 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $0.496 + 0.868i$
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: \((2,\ 33,\ (\ :2),\ 0.496 + 0.868i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.71629 - 0.996073i\)
\(L(\frac12)\) \(\approx\) \(1.71629 - 0.996073i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-7.81 + 4.46i)T \)
11 \( 1 - 36.4iT \)
good2 \( 1 + 2.36iT - 16T^{2} \)
5 \( 1 - 9.92iT - 625T^{2} \)
7 \( 1 + 13.3T + 2.40e3T^{2} \)
13 \( 1 + 138.T + 2.85e4T^{2} \)
17 \( 1 - 445. iT - 8.35e4T^{2} \)
19 \( 1 + 151.T + 1.30e5T^{2} \)
23 \( 1 - 661. iT - 2.79e5T^{2} \)
29 \( 1 + 130. iT - 7.07e5T^{2} \)
31 \( 1 + 1.03e3T + 9.23e5T^{2} \)
37 \( 1 - 1.11e3T + 1.87e6T^{2} \)
41 \( 1 + 1.27e3iT - 2.82e6T^{2} \)
43 \( 1 + 1.13e3T + 3.41e6T^{2} \)
47 \( 1 + 915. iT - 4.87e6T^{2} \)
53 \( 1 - 1.30e3iT - 7.89e6T^{2} \)
59 \( 1 + 1.57e3iT - 1.21e7T^{2} \)
61 \( 1 + 5.83e3T + 1.38e7T^{2} \)
67 \( 1 - 7.20e3T + 2.01e7T^{2} \)
71 \( 1 + 9.59e3iT - 2.54e7T^{2} \)
73 \( 1 - 1.05e4T + 2.83e7T^{2} \)
79 \( 1 - 1.91e3T + 3.89e7T^{2} \)
83 \( 1 + 1.08e4iT - 4.74e7T^{2} \)
89 \( 1 - 6.02e3iT - 6.27e7T^{2} \)
97 \( 1 + 199.T + 8.85e7T^{2} \)
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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

−15.39860919471426603540463692830, −14.66390370991988198037180473808, −13.06592583682663760712407863797, −12.23674065498288657578000022908, −10.71885008096959976289958021110, −9.492405049042558064845578939140, −7.71135506207232597796684314975, −6.52889092808864993222027639677, −3.49339954739542190199337775099, −1.95668260461259627631196430166, 2.69006146467603872202887313513, 4.95715528684772585448685479544, 6.93824499462606343664870455938, 8.248272537107441248074513775262, 9.558063129055871438770092463811, 11.04907554005211469054199086061, 12.65539264324210000192836442269, 14.16914254838973826319584341901, 15.00110970568927874534487315728, 16.21900284343242111935980012761

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