↑

Properties

Label 2-33-3.2-c8-0-5
Degree $2$
Conductor $33$
Sign $-0.198 + 0.980i$
Analytic cond. $13.4434$
Root an. cond. $3.66653$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 30.7i·2-s + (79.3 + 16.0i)3-s − 686.·4-s + 657. i·5-s + (−493. + 2.43e3i)6-s − 1.25e3·7-s − 1.32e4i·8-s + (6.04e3 + 2.55e3i)9-s − 2.01e4·10-s + 4.41e3i·11-s + (−5.45e4 − 1.10e4i)12-s − 4.00e4·13-s − 3.85e4i·14-s + (−1.05e4 + 5.21e4i)15-s + 2.30e5·16-s − 4.88e4i·17-s + ⋯
L(s)  = 1  + 1.91i·2-s + (0.980 + 0.198i)3-s − 2.68·4-s + 1.05i·5-s + (−0.380 + 1.88i)6-s − 0.522·7-s − 3.23i·8-s + (0.921 + 0.388i)9-s − 2.01·10-s + 0.301i·11-s + (−2.63 − 0.532i)12-s − 1.40·13-s − 1.00i·14-s + (−0.208 + 1.03i)15-s + 3.51·16-s − 0.584i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.198 + 0.980i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.198 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $-0.198 + 0.980i$
Analytic conductor: \(13.4434\)
Root analytic conductor: \(3.66653\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{33} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 33,\ (\ :4),\ -0.198 + 0.980i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.884884 - 1.08198i\)
\(L(\frac12)\) \(\approx\) \(0.884884 - 1.08198i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-79.3 - 16.0i)T \)
11 \( 1 - 4.41e3iT \)
good2 \( 1 - 30.7iT - 256T^{2} \)
5 \( 1 - 657. iT - 3.90e5T^{2} \)
7 \( 1 + 1.25e3T + 5.76e6T^{2} \)
13 \( 1 + 4.00e4T + 8.15e8T^{2} \)
17 \( 1 + 4.88e4iT - 6.97e9T^{2} \)
19 \( 1 - 1.43e5T + 1.69e10T^{2} \)
23 \( 1 - 1.27e5iT - 7.83e10T^{2} \)
29 \( 1 - 1.23e6iT - 5.00e11T^{2} \)
31 \( 1 + 1.23e6T + 8.52e11T^{2} \)
37 \( 1 - 4.93e5T + 3.51e12T^{2} \)
41 \( 1 + 2.58e6iT - 7.98e12T^{2} \)
43 \( 1 + 4.87e6T + 1.16e13T^{2} \)
47 \( 1 - 3.99e6iT - 2.38e13T^{2} \)
53 \( 1 - 7.34e6iT - 6.22e13T^{2} \)
59 \( 1 - 1.78e6iT - 1.46e14T^{2} \)
61 \( 1 - 8.49e6T + 1.91e14T^{2} \)
67 \( 1 - 2.49e6T + 4.06e14T^{2} \)
71 \( 1 - 4.22e7iT - 6.45e14T^{2} \)
73 \( 1 + 9.54e6T + 8.06e14T^{2} \)
79 \( 1 - 5.39e7T + 1.51e15T^{2} \)
83 \( 1 - 7.19e6iT - 2.25e15T^{2} \)
89 \( 1 + 4.49e7iT - 3.93e15T^{2} \)
97 \( 1 - 1.04e8T + 7.83e15T^{2} \)
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

−15.59652508950247214658174097576, −14.62323887861419764716863011406, −14.15476491993459032324528130971, −12.83448663658821626738173907852, −10.00444548912668239914372132529, −9.135967967357233821208105843915, −7.46031794826789668534891949692, −6.99895092468996385116136420093, −5.07731957072148760857017207991, −3.29218434473882363670482875058, 0.52990227710366662750441096945, 2.01420857269345646771542517224, 3.40861267687434282103067991078, 4.82038975758198179619748576439, 8.118489083477856709644599800532, 9.310340226090364596969448349683, 9.984602058272973769422535301596, 11.83182600830777185422306802760, 12.76162298911999811662657767672, 13.43515843281697280069279565586

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