↑

Properties

Label 2-33-11.2-c2-0-0
Degree $2$
Conductor $33$
Sign $0.315 - 0.949i$
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  + (−3.47 − 1.12i)2-s + (−1.40 + 1.01i)3-s + (7.54 + 5.48i)4-s + (1.69 + 5.20i)5-s + (6.01 − 1.95i)6-s + (−4.20 + 5.79i)7-s + (−11.4 − 15.7i)8-s + (0.927 − 2.85i)9-s − 19.9i·10-s + (0.170 + 10.9i)11-s − 16.1·12-s + (−6.27 − 2.03i)13-s + (21.1 − 15.3i)14-s + (−7.66 − 5.56i)15-s + (10.4 + 32.0i)16-s + (17.5 − 5.70i)17-s + ⋯
L(s)  = 1  + (−1.73 − 0.564i)2-s + (−0.467 + 0.339i)3-s + (1.88 + 1.37i)4-s + (0.338 + 1.04i)5-s + (1.00 − 0.325i)6-s + (−0.601 + 0.827i)7-s + (−1.43 − 1.96i)8-s + (0.103 − 0.317i)9-s − 1.99i·10-s + (0.0155 + 0.999i)11-s − 1.34·12-s + (−0.482 − 0.156i)13-s + (1.51 − 1.09i)14-s + (−0.510 − 0.371i)15-s + (0.651 + 2.00i)16-s + (1.03 − 0.335i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.315 - 0.949i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.315 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $0.315 - 0.949i$
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: \((2,\ 33,\ (\ :1),\ 0.315 - 0.949i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.313008 + 0.225877i\)
\(L(\frac12)\) \(\approx\) \(0.313008 + 0.225877i\)
\(L(2)\) 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 \( 1 + (-0.170 - 10.9i)T \)
good2 \( 1 + (3.47 + 1.12i)T + (3.23 + 2.35i)T^{2} \)
5 \( 1 + (-1.69 - 5.20i)T + (-20.2 + 14.6i)T^{2} \)
7 \( 1 + (4.20 - 5.79i)T + (-15.1 - 46.6i)T^{2} \)
13 \( 1 + (6.27 + 2.03i)T + (136. + 99.3i)T^{2} \)
17 \( 1 + (-17.5 + 5.70i)T + (233. - 169. i)T^{2} \)
19 \( 1 + (-4.95 - 6.81i)T + (-111. + 343. i)T^{2} \)
23 \( 1 + 17.7T + 529T^{2} \)
29 \( 1 + (-11.8 + 16.2i)T + (-259. - 799. i)T^{2} \)
31 \( 1 + (-10.9 + 33.6i)T + (-777. - 564. i)T^{2} \)
37 \( 1 + (-39.4 - 28.6i)T + (423. + 1.30e3i)T^{2} \)
41 \( 1 + (-18.5 - 25.5i)T + (-519. + 1.59e3i)T^{2} \)
43 \( 1 - 45.0iT - 1.84e3T^{2} \)
47 \( 1 + (-0.589 + 0.428i)T + (682. - 2.10e3i)T^{2} \)
53 \( 1 + (-21.6 + 66.7i)T + (-2.27e3 - 1.65e3i)T^{2} \)
59 \( 1 + (19.5 + 14.1i)T + (1.07e3 + 3.31e3i)T^{2} \)
61 \( 1 + (39.0 - 12.6i)T + (3.01e3 - 2.18e3i)T^{2} \)
67 \( 1 - 96.0T + 4.48e3T^{2} \)
71 \( 1 + (-12.2 - 37.6i)T + (-4.07e3 + 2.96e3i)T^{2} \)
73 \( 1 + (-41.2 + 56.8i)T + (-1.64e3 - 5.06e3i)T^{2} \)
79 \( 1 + (84.9 + 27.6i)T + (5.04e3 + 3.66e3i)T^{2} \)
83 \( 1 + (-24.9 + 8.10i)T + (5.57e3 - 4.04e3i)T^{2} \)
89 \( 1 + 118.T + 7.92e3T^{2} \)
97 \( 1 + (10.2 - 31.4i)T + (-7.61e3 - 5.53e3i)T^{2} \)
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

−17.08587147334140500909455174962, −15.97686328555703170395955833827, −14.82248081932124127344465721342, −12.39417730987202063553207069677, −11.45256329279713499435311963833, −9.934041699317255565791946635332, −9.758953714222189437933956500733, −7.77228904791527693651112639293, −6.34338071303679713674744474805, −2.67195670271665527932868624657, 0.884389385998805316190511189226, 5.76006052594412553921182629560, 7.17643127493752518343820672191, 8.486350848253496982621768208710, 9.700215648414016257873231915866, 10.78263759678213833938898433487, 12.37716845357227688455314188305, 13.96597249242210643084772560504, 15.94309089429304071021479229305, 16.66493756587703796334045200649

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