Properties

Label 2-33-11.4-c5-0-0
Degree $2$
Conductor $33$
Sign $0.998 - 0.0456i$
Analytic cond. $5.29266$
Root an. cond. $2.30057$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−7.93 − 5.76i)2-s + (2.78 − 8.55i)3-s + (19.8 + 61.0i)4-s + (−37.2 + 27.0i)5-s + (−71.4 + 51.8i)6-s + (37.2 + 114. i)7-s + (97.6 − 300. i)8-s + (−65.5 − 47.6i)9-s + 451.·10-s + (319. − 242. i)11-s + 578.·12-s + (488. + 354. i)13-s + (365. − 1.12e3i)14-s + (128. + 394. i)15-s + (−845. + 614. i)16-s + (−1.12e3 + 817. i)17-s + ⋯
L(s)  = 1  + (−1.40 − 1.01i)2-s + (0.178 − 0.549i)3-s + (0.620 + 1.90i)4-s + (−0.666 + 0.484i)5-s + (−0.809 + 0.588i)6-s + (0.287 + 0.884i)7-s + (0.539 − 1.66i)8-s + (−0.269 − 0.195i)9-s + 1.42·10-s + (0.795 − 0.605i)11-s + 1.15·12-s + (0.801 + 0.582i)13-s + (0.498 − 1.53i)14-s + (0.147 + 0.452i)15-s + (−0.826 + 0.600i)16-s + (−0.944 + 0.686i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0456i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0456i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $0.998 - 0.0456i$
Analytic conductor: \(5.29266\)
Root analytic conductor: \(2.30057\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{33} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 33,\ (\ :5/2),\ 0.998 - 0.0456i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.662895 + 0.0151383i\)
\(L(\frac12)\) \(\approx\) \(0.662895 + 0.0151383i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.78 + 8.55i)T \)
11 \( 1 + (-319. + 242. i)T \)
good2 \( 1 + (7.93 + 5.76i)T + (9.88 + 30.4i)T^{2} \)
5 \( 1 + (37.2 - 27.0i)T + (965. - 2.97e3i)T^{2} \)
7 \( 1 + (-37.2 - 114. i)T + (-1.35e4 + 9.87e3i)T^{2} \)
13 \( 1 + (-488. - 354. i)T + (1.14e5 + 3.53e5i)T^{2} \)
17 \( 1 + (1.12e3 - 817. i)T + (4.38e5 - 1.35e6i)T^{2} \)
19 \( 1 + (258. - 794. i)T + (-2.00e6 - 1.45e6i)T^{2} \)
23 \( 1 - 3.26e3T + 6.43e6T^{2} \)
29 \( 1 + (-2.61e3 - 8.04e3i)T + (-1.65e7 + 1.20e7i)T^{2} \)
31 \( 1 + (-5.17e3 - 3.76e3i)T + (8.84e6 + 2.72e7i)T^{2} \)
37 \( 1 + (3.63e3 + 1.11e4i)T + (-5.61e7 + 4.07e7i)T^{2} \)
41 \( 1 + (3.29e3 - 1.01e4i)T + (-9.37e7 - 6.80e7i)T^{2} \)
43 \( 1 + 2.64e3T + 1.47e8T^{2} \)
47 \( 1 + (4.45e3 - 1.37e4i)T + (-1.85e8 - 1.34e8i)T^{2} \)
53 \( 1 + (3.20e4 + 2.32e4i)T + (1.29e8 + 3.97e8i)T^{2} \)
59 \( 1 + (-3.12e3 - 9.61e3i)T + (-5.78e8 + 4.20e8i)T^{2} \)
61 \( 1 + (-2.12e4 + 1.54e4i)T + (2.60e8 - 8.03e8i)T^{2} \)
67 \( 1 + 1.06e4T + 1.35e9T^{2} \)
71 \( 1 + (-6.33e3 + 4.60e3i)T + (5.57e8 - 1.71e9i)T^{2} \)
73 \( 1 + (-1.35e4 - 4.15e4i)T + (-1.67e9 + 1.21e9i)T^{2} \)
79 \( 1 + (5.28e4 + 3.84e4i)T + (9.50e8 + 2.92e9i)T^{2} \)
83 \( 1 + (1.35e4 - 9.81e3i)T + (1.21e9 - 3.74e9i)T^{2} \)
89 \( 1 - 5.53e4T + 5.58e9T^{2} \)
97 \( 1 + (2.34e4 + 1.70e4i)T + (2.65e9 + 8.16e9i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−16.02435704992927326209699628051, −14.52765967393908474214167765937, −12.69635979380368804050026922639, −11.55236815269070896121634973738, −10.92305319791342251159422601824, −8.994574626193827419881459168534, −8.389612327763237939572069538856, −6.74787029873784952244195016926, −3.27778754906266652264062745880, −1.51146708257951135248596827811, 0.68979503476020814811808867932, 4.49221091101285667318668370421, 6.67528932816705848316726403527, 7.977354590354724074640228307465, 8.972896491513527342574577586722, 10.19487180236731243134753599583, 11.43896005210422613694885541542, 13.65780412729542088205849844424, 15.24111466540558429232542852696, 15.71850531908973184117274361252

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