↑

Properties

Label 2-33-11.2-c2-0-2
Degree $2$
Conductor $33$
Sign $-0.105 + 0.994i$
Analytic cond. $0.899184$
Root an. cond. $0.948253$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−2.40 − 0.782i)2-s + (1.40 − 1.01i)3-s + (1.94 + 1.41i)4-s + (−2.61 − 8.03i)5-s + (−4.17 + 1.35i)6-s + (1.43 − 1.97i)7-s + (2.36 + 3.25i)8-s + (0.927 − 2.85i)9-s + 21.3i·10-s + (2.51 + 10.7i)11-s + 4.17·12-s + (11.1 + 3.63i)13-s + (−4.99 + 3.63i)14-s + (−11.8 − 8.59i)15-s + (−6.12 − 18.8i)16-s + (−1.93 + 0.627i)17-s + ⋯
L(s)  = 1  + (−1.20 − 0.391i)2-s + (0.467 − 0.339i)3-s + (0.487 + 0.354i)4-s + (−0.522 − 1.60i)5-s + (−0.695 + 0.225i)6-s + (0.204 − 0.282i)7-s + (0.295 + 0.407i)8-s + (0.103 − 0.317i)9-s + 2.13i·10-s + (0.228 + 0.973i)11-s + 0.347·12-s + (0.861 + 0.279i)13-s + (−0.357 + 0.259i)14-s + (−0.789 − 0.573i)15-s + (−0.383 − 1.17i)16-s + (−0.113 + 0.0368i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.105 + 0.994i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.105 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $-0.105 + 0.994i$
Analytic conductor: \(0.899184\)
Root analytic conductor: \(0.948253\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{33} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 33,\ (\ :1),\ -0.105 + 0.994i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.413594 - 0.459600i\)
\(L(\frac12)\) \(\approx\) \(0.413594 - 0.459600i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.40 + 1.01i)T \)
11 \( 1 + (-2.51 - 10.7i)T \)
good2 \( 1 + (2.40 + 0.782i)T + (3.23 + 2.35i)T^{2} \)
5 \( 1 + (2.61 + 8.03i)T + (-20.2 + 14.6i)T^{2} \)
7 \( 1 + (-1.43 + 1.97i)T + (-15.1 - 46.6i)T^{2} \)
13 \( 1 + (-11.1 - 3.63i)T + (136. + 99.3i)T^{2} \)
17 \( 1 + (1.93 - 0.627i)T + (233. - 169. i)T^{2} \)
19 \( 1 + (4.97 + 6.85i)T + (-111. + 343. i)T^{2} \)
23 \( 1 - 41.9T + 529T^{2} \)
29 \( 1 + (14.4 - 19.9i)T + (-259. - 799. i)T^{2} \)
31 \( 1 + (-6.74 + 20.7i)T + (-777. - 564. i)T^{2} \)
37 \( 1 + (-12.9 - 9.40i)T + (423. + 1.30e3i)T^{2} \)
41 \( 1 + (30.9 + 42.5i)T + (-519. + 1.59e3i)T^{2} \)
43 \( 1 - 42.3iT - 1.84e3T^{2} \)
47 \( 1 + (-13.0 + 9.51i)T + (682. - 2.10e3i)T^{2} \)
53 \( 1 + (15.3 - 47.1i)T + (-2.27e3 - 1.65e3i)T^{2} \)
59 \( 1 + (16.1 + 11.7i)T + (1.07e3 + 3.31e3i)T^{2} \)
61 \( 1 + (113. - 36.7i)T + (3.01e3 - 2.18e3i)T^{2} \)
67 \( 1 + 4.41T + 4.48e3T^{2} \)
71 \( 1 + (-1.86 - 5.74i)T + (-4.07e3 + 2.96e3i)T^{2} \)
73 \( 1 + (3.57 - 4.91i)T + (-1.64e3 - 5.06e3i)T^{2} \)
79 \( 1 + (-98.3 - 31.9i)T + (5.04e3 + 3.66e3i)T^{2} \)
83 \( 1 + (-28.6 + 9.32i)T + (5.57e3 - 4.04e3i)T^{2} \)
89 \( 1 + 60.4T + 7.92e3T^{2} \)
97 \( 1 + (-11.3 + 34.8i)T + (-7.61e3 - 5.53e3i)T^{2} \)
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

−16.66045737226041607259969431564, −15.24415790921424725646956984921, −13.52211206071612863231682282298, −12.41268872315182130711006185247, −11.07323044806379697275739220982, −9.321288358928538965997047377259, −8.691068542421153119849660564467, −7.48133186062588208754430263441, −4.63787193544416285292111844226, −1.28812877788889838587662610745, 3.41865471932383632168954996753, 6.55456962059473806062355306729, 7.86801333159947138353060273685, 8.944513454662689650141962867198, 10.47054100607757089200781975763, 11.22863232450610247123633245393, 13.53050533899430866783332868768, 14.86951699008092181540681324182, 15.64551994738660970902493222035, 16.87144330349527370258715040106

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