Properties

Label 2-33-11.8-c4-0-7
Degree $2$
Conductor $33$
Sign $0.0963 - 0.995i$
Analytic cond. $3.41120$
Root an. cond. $1.84694$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−4.37 − 6.01i)2-s + (−1.60 − 4.94i)3-s + (−12.1 + 37.3i)4-s + (−24.8 − 18.0i)5-s + (−22.7 + 31.2i)6-s + (40.6 + 13.1i)7-s + (164. − 53.5i)8-s + (−21.8 + 15.8i)9-s + 228. i·10-s + (−55.8 + 107. i)11-s + 204.·12-s + (−92.9 − 127. i)13-s + (−98.1 − 302. i)14-s + (−49.4 + 152. i)15-s + (−534. − 388. i)16-s + (−52.6 + 72.4i)17-s + ⋯
L(s)  = 1  + (−1.09 − 1.50i)2-s + (−0.178 − 0.549i)3-s + (−0.759 + 2.33i)4-s + (−0.995 − 0.723i)5-s + (−0.630 + 0.868i)6-s + (0.828 + 0.269i)7-s + (2.57 − 0.836i)8-s + (−0.269 + 0.195i)9-s + 2.28i·10-s + (−0.461 + 0.886i)11-s + 1.41·12-s + (−0.549 − 0.756i)13-s + (−0.500 − 1.54i)14-s + (−0.219 + 0.675i)15-s + (−2.08 − 1.51i)16-s + (−0.182 + 0.250i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0963 - 0.995i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.0963 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $0.0963 - 0.995i$
Analytic conductor: \(3.41120\)
Root analytic conductor: \(1.84694\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{33} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 33,\ (\ :2),\ 0.0963 - 0.995i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.0991556 + 0.0900251i\)
\(L(\frac12)\) \(\approx\) \(0.0991556 + 0.0900251i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.60 + 4.94i)T \)
11 \( 1 + (55.8 - 107. i)T \)
good2 \( 1 + (4.37 + 6.01i)T + (-4.94 + 15.2i)T^{2} \)
5 \( 1 + (24.8 + 18.0i)T + (193. + 594. i)T^{2} \)
7 \( 1 + (-40.6 - 13.1i)T + (1.94e3 + 1.41e3i)T^{2} \)
13 \( 1 + (92.9 + 127. i)T + (-8.82e3 + 2.71e4i)T^{2} \)
17 \( 1 + (52.6 - 72.4i)T + (-2.58e4 - 7.94e4i)T^{2} \)
19 \( 1 + (624. - 202. i)T + (1.05e5 - 7.66e4i)T^{2} \)
23 \( 1 + 292.T + 2.79e5T^{2} \)
29 \( 1 + (646. + 210. i)T + (5.72e5 + 4.15e5i)T^{2} \)
31 \( 1 + (-1.42e3 + 1.03e3i)T + (2.85e5 - 8.78e5i)T^{2} \)
37 \( 1 + (-56.5 + 173. i)T + (-1.51e6 - 1.10e6i)T^{2} \)
41 \( 1 + (553. - 179. i)T + (2.28e6 - 1.66e6i)T^{2} \)
43 \( 1 - 298. iT - 3.41e6T^{2} \)
47 \( 1 + (662. + 2.03e3i)T + (-3.94e6 + 2.86e6i)T^{2} \)
53 \( 1 + (1.78e3 - 1.29e3i)T + (2.43e6 - 7.50e6i)T^{2} \)
59 \( 1 + (-1.49e3 + 4.60e3i)T + (-9.80e6 - 7.12e6i)T^{2} \)
61 \( 1 + (1.17e3 - 1.62e3i)T + (-4.27e6 - 1.31e7i)T^{2} \)
67 \( 1 + 4.98e3T + 2.01e7T^{2} \)
71 \( 1 + (-4.37e3 - 3.18e3i)T + (7.85e6 + 2.41e7i)T^{2} \)
73 \( 1 + (-1.92e3 - 625. i)T + (2.29e7 + 1.66e7i)T^{2} \)
79 \( 1 + (1.82e3 + 2.50e3i)T + (-1.20e7 + 3.70e7i)T^{2} \)
83 \( 1 + (3.44e3 - 4.73e3i)T + (-1.46e7 - 4.51e7i)T^{2} \)
89 \( 1 + 1.48e3T + 6.27e7T^{2} \)
97 \( 1 + (-5.29e3 + 3.84e3i)T + (2.73e7 - 8.41e7i)T^{2} \)
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

−15.18732680264862331779919050193, −12.93032587186319496426911088977, −12.27829715763966286900908492771, −11.36873419494683083393459674813, −10.10029353429548187414331657928, −8.368569476981832697628219030612, −7.85566083419686404893427603650, −4.39763184953176604380619893845, −2.05275430161302512057049190670, −0.13747566647020761975287223260, 4.67378024340112557519803007500, 6.50571415645607614060786698750, 7.76624946830354627190202645224, 8.778759391805505368492165322317, 10.43116786843935258322576197435, 11.31715221931061214775950620411, 14.08893762646893242749764179487, 14.96313021945279967504152570453, 15.76144645767398833489838477689, 16.76312293028636974613898721708

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