Properties

Label 2-33-11.4-c3-0-4
Degree $2$
Conductor $33$
Sign $0.731 + 0.682i$
Analytic cond. $1.94706$
Root an. cond. $1.39537$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.523 + 0.380i)2-s + (0.927 − 2.85i)3-s + (−2.34 − 7.21i)4-s + (9.01 − 6.54i)5-s + (1.57 − 1.14i)6-s + (8.07 + 24.8i)7-s + (3.11 − 9.58i)8-s + (−7.28 − 5.29i)9-s + 7.20·10-s + (−36.0 − 5.31i)11-s − 22.7·12-s + (43.2 + 31.4i)13-s + (−5.22 + 16.0i)14-s + (−10.3 − 31.7i)15-s + (−43.7 + 31.8i)16-s + (−18.9 + 13.7i)17-s + ⋯
L(s)  = 1  + (0.185 + 0.134i)2-s + (0.178 − 0.549i)3-s + (−0.292 − 0.901i)4-s + (0.806 − 0.585i)5-s + (0.106 − 0.0776i)6-s + (0.436 + 1.34i)7-s + (0.137 − 0.423i)8-s + (−0.269 − 0.195i)9-s + 0.227·10-s + (−0.989 − 0.145i)11-s − 0.547·12-s + (0.923 + 0.671i)13-s + (−0.0997 + 0.307i)14-s + (−0.177 − 0.547i)15-s + (−0.684 + 0.497i)16-s + (−0.270 + 0.196i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.731 + 0.682i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.731 + 0.682i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $0.731 + 0.682i$
Analytic conductor: \(1.94706\)
Root analytic conductor: \(1.39537\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{33} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 33,\ (\ :3/2),\ 0.731 + 0.682i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.36194 - 0.536682i\)
\(L(\frac12)\) \(\approx\) \(1.36194 - 0.536682i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.927 + 2.85i)T \)
11 \( 1 + (36.0 + 5.31i)T \)
good2 \( 1 + (-0.523 - 0.380i)T + (2.47 + 7.60i)T^{2} \)
5 \( 1 + (-9.01 + 6.54i)T + (38.6 - 118. i)T^{2} \)
7 \( 1 + (-8.07 - 24.8i)T + (-277. + 201. i)T^{2} \)
13 \( 1 + (-43.2 - 31.4i)T + (678. + 2.08e3i)T^{2} \)
17 \( 1 + (18.9 - 13.7i)T + (1.51e3 - 4.67e3i)T^{2} \)
19 \( 1 + (21.8 - 67.3i)T + (-5.54e3 - 4.03e3i)T^{2} \)
23 \( 1 - 164.T + 1.21e4T^{2} \)
29 \( 1 + (67.5 + 207. i)T + (-1.97e4 + 1.43e4i)T^{2} \)
31 \( 1 + (-62.0 - 45.0i)T + (9.20e3 + 2.83e4i)T^{2} \)
37 \( 1 + (87.5 + 269. i)T + (-4.09e4 + 2.97e4i)T^{2} \)
41 \( 1 + (-1.50 + 4.61i)T + (-5.57e4 - 4.05e4i)T^{2} \)
43 \( 1 + 333.T + 7.95e4T^{2} \)
47 \( 1 + (121. - 374. i)T + (-8.39e4 - 6.10e4i)T^{2} \)
53 \( 1 + (123. + 89.3i)T + (4.60e4 + 1.41e5i)T^{2} \)
59 \( 1 + (237. + 729. i)T + (-1.66e5 + 1.20e5i)T^{2} \)
61 \( 1 + (-287. + 209. i)T + (7.01e4 - 2.15e5i)T^{2} \)
67 \( 1 - 102.T + 3.00e5T^{2} \)
71 \( 1 + (504. - 366. i)T + (1.10e5 - 3.40e5i)T^{2} \)
73 \( 1 + (95.7 + 294. i)T + (-3.14e5 + 2.28e5i)T^{2} \)
79 \( 1 + (-517. - 375. i)T + (1.52e5 + 4.68e5i)T^{2} \)
83 \( 1 + (233. - 169. i)T + (1.76e5 - 5.43e5i)T^{2} \)
89 \( 1 - 184.T + 7.04e5T^{2} \)
97 \( 1 + (515. + 374. i)T + (2.82e5 + 8.68e5i)T^{2} \)
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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

−15.85260845874832216201291189155, −14.80395367765084848088116569935, −13.61125533731548818343515555381, −12.77982084131360044009417679476, −11.13165208443043039062678621362, −9.454768517499498743535875657801, −8.468985607281277290593730468897, −6.16271692677866093089773720631, −5.21118236109139042911201762805, −1.83653963426987260236137631995, 3.13410729031416747220649615240, 4.85075632742316730143632564392, 7.14566249398237995335834626232, 8.575738439258230534627506281387, 10.29990452358594940538775087303, 11.09332218227775437639427189497, 13.22995059942401453618238054285, 13.63838058274282662095092514258, 15.08452290435747457822310249648, 16.54350434566011780511709011895

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