Properties

Label 2-33-11.6-c4-0-0
Degree $2$
Conductor $33$
Sign $-0.514 - 0.857i$
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  + (0.743 − 0.241i)2-s + (−4.20 − 3.05i)3-s + (−12.4 + 9.04i)4-s + (−9.71 + 29.9i)5-s + (−3.86 − 1.25i)6-s + (16.8 + 23.2i)7-s + (−14.4 + 19.8i)8-s + (8.34 + 25.6i)9-s + 24.5i·10-s + (−24.4 − 118. i)11-s + 79.9·12-s + (−294. + 95.8i)13-s + (18.1 + 13.1i)14-s + (132. − 96.0i)15-s + (70.1 − 215. i)16-s + (397. + 129. i)17-s + ⋯
L(s)  = 1  + (0.185 − 0.0603i)2-s + (−0.467 − 0.339i)3-s + (−0.778 + 0.565i)4-s + (−0.388 + 1.19i)5-s + (−0.107 − 0.0348i)6-s + (0.344 + 0.474i)7-s + (−0.225 + 0.310i)8-s + (0.103 + 0.317i)9-s + 0.245i·10-s + (−0.202 − 0.979i)11-s + 0.555·12-s + (−1.74 + 0.567i)13-s + (0.0926 + 0.0673i)14-s + (0.587 − 0.426i)15-s + (0.274 − 0.843i)16-s + (1.37 + 0.446i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.514 - 0.857i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.514 - 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $-0.514 - 0.857i$
Analytic conductor: \(3.41120\)
Root analytic conductor: \(1.84694\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{33} (28, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 33,\ (\ :2),\ -0.514 - 0.857i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.364545 + 0.643518i\)
\(L(\frac12)\) \(\approx\) \(0.364545 + 0.643518i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (4.20 + 3.05i)T \)
11 \( 1 + (24.4 + 118. i)T \)
good2 \( 1 + (-0.743 + 0.241i)T + (12.9 - 9.40i)T^{2} \)
5 \( 1 + (9.71 - 29.9i)T + (-505. - 367. i)T^{2} \)
7 \( 1 + (-16.8 - 23.2i)T + (-741. + 2.28e3i)T^{2} \)
13 \( 1 + (294. - 95.8i)T + (2.31e4 - 1.67e4i)T^{2} \)
17 \( 1 + (-397. - 129. i)T + (6.75e4 + 4.90e4i)T^{2} \)
19 \( 1 + (144. - 198. i)T + (-4.02e4 - 1.23e5i)T^{2} \)
23 \( 1 - 529.T + 2.79e5T^{2} \)
29 \( 1 + (-164. - 226. i)T + (-2.18e5 + 6.72e5i)T^{2} \)
31 \( 1 + (-443. - 1.36e3i)T + (-7.47e5 + 5.42e5i)T^{2} \)
37 \( 1 + (1.46e3 - 1.06e3i)T + (5.79e5 - 1.78e6i)T^{2} \)
41 \( 1 + (289. - 398. i)T + (-8.73e5 - 2.68e6i)T^{2} \)
43 \( 1 + 287. iT - 3.41e6T^{2} \)
47 \( 1 + (369. + 268. i)T + (1.50e6 + 4.64e6i)T^{2} \)
53 \( 1 + (343. + 1.05e3i)T + (-6.38e6 + 4.63e6i)T^{2} \)
59 \( 1 + (295. - 214. i)T + (3.74e6 - 1.15e7i)T^{2} \)
61 \( 1 + (-445. - 144. i)T + (1.12e7 + 8.13e6i)T^{2} \)
67 \( 1 - 5.31e3T + 2.01e7T^{2} \)
71 \( 1 + (2.93e3 - 9.04e3i)T + (-2.05e7 - 1.49e7i)T^{2} \)
73 \( 1 + (2.51e3 + 3.46e3i)T + (-8.77e6 + 2.70e7i)T^{2} \)
79 \( 1 + (6.20e3 - 2.01e3i)T + (3.15e7 - 2.28e7i)T^{2} \)
83 \( 1 + (-5.63e3 - 1.83e3i)T + (3.83e7 + 2.78e7i)T^{2} \)
89 \( 1 - 5.36e3T + 6.27e7T^{2} \)
97 \( 1 + (-1.24e3 - 3.84e3i)T + (-7.16e7 + 5.20e7i)T^{2} \)
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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

−16.61606938762994911812844079361, −14.81515537860651946043668627160, −14.10665941468955516893307361878, −12.49323202856523311159378319702, −11.66518366176024265381818493102, −10.22171889000362907066465733204, −8.367741932648158524828085131229, −7.07109235744302897027326648432, −5.19556907446692172491711663058, −3.13114127091501939506346088270, 0.55203782018277697319205938013, 4.55455221449560100271389528986, 5.19856206475509559661995302491, 7.60628557719503097262443115163, 9.311500885139189851571434475916, 10.26212234599596019715843134512, 12.11877560492284030750984510762, 12.93667472257351693888355572260, 14.50782141517929288459065847162, 15.44810900208204639935284389138

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