Properties

Label 2-33-11.3-c3-0-3
Degree $2$
Conductor $33$
Sign $-0.299 + 0.954i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−2.73 + 1.98i)2-s + (−0.927 − 2.85i)3-s + (1.05 − 3.23i)4-s + (−11.3 − 8.23i)5-s + (8.19 + 5.95i)6-s + (7.21 − 22.1i)7-s + (−4.79 − 14.7i)8-s + (−7.28 + 5.29i)9-s + 47.2·10-s + (−28.7 + 22.4i)11-s − 10.2·12-s + (−30.5 + 22.1i)13-s + (24.3 + 74.9i)14-s + (−12.9 + 39.9i)15-s + (64.4 + 46.8i)16-s + (−6.96 − 5.05i)17-s + ⋯
L(s)  = 1  + (−0.965 + 0.701i)2-s + (−0.178 − 0.549i)3-s + (0.131 − 0.404i)4-s + (−1.01 − 0.736i)5-s + (0.557 + 0.405i)6-s + (0.389 − 1.19i)7-s + (−0.211 − 0.652i)8-s + (−0.269 + 0.195i)9-s + 1.49·10-s + (−0.788 + 0.615i)11-s − 0.245·12-s + (−0.651 + 0.473i)13-s + (0.465 + 1.43i)14-s + (−0.223 + 0.687i)15-s + (1.00 + 0.731i)16-s + (−0.0993 − 0.0721i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.299 + 0.954i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.299 + 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $-0.299 + 0.954i$
Analytic conductor: \(1.94706\)
Root analytic conductor: \(1.39537\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{33} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 33,\ (\ :3/2),\ -0.299 + 0.954i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.206554 - 0.281230i\)
\(L(\frac12)\) \(\approx\) \(0.206554 - 0.281230i\)
\(L(\frac{5}{2})\) 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 \( 1 + (28.7 - 22.4i)T \)
good2 \( 1 + (2.73 - 1.98i)T + (2.47 - 7.60i)T^{2} \)
5 \( 1 + (11.3 + 8.23i)T + (38.6 + 118. i)T^{2} \)
7 \( 1 + (-7.21 + 22.1i)T + (-277. - 201. i)T^{2} \)
13 \( 1 + (30.5 - 22.1i)T + (678. - 2.08e3i)T^{2} \)
17 \( 1 + (6.96 + 5.05i)T + (1.51e3 + 4.67e3i)T^{2} \)
19 \( 1 + (27.3 + 84.2i)T + (-5.54e3 + 4.03e3i)T^{2} \)
23 \( 1 - 137.T + 1.21e4T^{2} \)
29 \( 1 + (-68.7 + 211. i)T + (-1.97e4 - 1.43e4i)T^{2} \)
31 \( 1 + (195. - 141. i)T + (9.20e3 - 2.83e4i)T^{2} \)
37 \( 1 + (-46.5 + 143. i)T + (-4.09e4 - 2.97e4i)T^{2} \)
41 \( 1 + (43.7 + 134. i)T + (-5.57e4 + 4.05e4i)T^{2} \)
43 \( 1 - 246.T + 7.95e4T^{2} \)
47 \( 1 + (-19.2 - 59.1i)T + (-8.39e4 + 6.10e4i)T^{2} \)
53 \( 1 + (-402. + 292. i)T + (4.60e4 - 1.41e5i)T^{2} \)
59 \( 1 + (181. - 557. i)T + (-1.66e5 - 1.20e5i)T^{2} \)
61 \( 1 + (523. + 380. i)T + (7.01e4 + 2.15e5i)T^{2} \)
67 \( 1 + 963.T + 3.00e5T^{2} \)
71 \( 1 + (-121. - 88.0i)T + (1.10e5 + 3.40e5i)T^{2} \)
73 \( 1 + (-133. + 411. i)T + (-3.14e5 - 2.28e5i)T^{2} \)
79 \( 1 + (-1.00e3 + 729. i)T + (1.52e5 - 4.68e5i)T^{2} \)
83 \( 1 + (577. + 419. i)T + (1.76e5 + 5.43e5i)T^{2} \)
89 \( 1 - 224.T + 7.04e5T^{2} \)
97 \( 1 + (-979. + 711. i)T + (2.82e5 - 8.68e5i)T^{2} \)
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

−16.23024178289223674897371464691, −15.10384347152971845065543222632, −13.32165678071662843738436433142, −12.23247765884733213906834698288, −10.70784750413652637633117415017, −9.012241704772395832088105879526, −7.68772673773453127456733609442, −7.14072630688227650613814971631, −4.52886337626886367500816656837, −0.43282054496099308948135698084, 2.90834818557474824855845871293, 5.42368404494586509073436703633, 7.87502079100337170728197820867, 8.997870916602898498415223680368, 10.51385755030383800823091846847, 11.23769658898422907585834173898, 12.31881093625013629583724166967, 14.71545970713354721472592188002, 15.29396568141350774305455530460, 16.71972375980616049787871784906

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