Properties

Label 2-30e2-4.3-c2-0-27
Degree $2$
Conductor $900$
Sign $-0.5 - 0.866i$
Analytic cond. $24.5232$
Root an. cond. $4.95209$
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  + (1.73 + i)2-s + (1.99 + 3.46i)4-s − 9.38i·7-s + 7.99i·8-s + 16.2i·11-s − 16.2·13-s + (9.38 − 16.2i)14-s + (−8 + 13.8i)16-s + 10.3·17-s + 20.7i·19-s + (−16.2 + 28.1i)22-s + 28i·23-s + (−28.1 − 16.2i)26-s + (32.4 − 18.7i)28-s + 9.38·29-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s − 1.34i·7-s + 0.999i·8-s + 1.47i·11-s − 1.24·13-s + (0.670 − 1.16i)14-s + (−0.5 + 0.866i)16-s + 0.611·17-s + 1.09i·19-s + (−0.738 + 1.27i)22-s + 1.21i·23-s + (−1.08 − 0.624i)26-s + (1.16 − 0.670i)28-s + 0.323·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.5 - 0.866i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.5 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.5 - 0.866i$
Analytic conductor: \(24.5232\)
Root analytic conductor: \(4.95209\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1),\ -0.5 - 0.866i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.595561175\)
\(L(\frac12)\) \(\approx\) \(2.595561175\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.73 - i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + 9.38iT - 49T^{2} \)
11 \( 1 - 16.2iT - 121T^{2} \)
13 \( 1 + 16.2T + 169T^{2} \)
17 \( 1 - 10.3T + 289T^{2} \)
19 \( 1 - 20.7iT - 361T^{2} \)
23 \( 1 - 28iT - 529T^{2} \)
29 \( 1 - 9.38T + 841T^{2} \)
31 \( 1 - 34.6iT - 961T^{2} \)
37 \( 1 - 48.7T + 1.36e3T^{2} \)
41 \( 1 - 18.7T + 1.68e3T^{2} \)
43 \( 1 - 37.5iT - 1.84e3T^{2} \)
47 \( 1 + 4iT - 2.20e3T^{2} \)
53 \( 1 - 31.1T + 2.80e3T^{2} \)
59 \( 1 - 16.2iT - 3.48e3T^{2} \)
61 \( 1 + 58T + 3.72e3T^{2} \)
67 \( 1 - 18.7iT - 4.48e3T^{2} \)
71 \( 1 + 97.4iT - 5.04e3T^{2} \)
73 \( 1 + 5.32e3T^{2} \)
79 \( 1 + 6.92iT - 6.24e3T^{2} \)
83 \( 1 + 32iT - 6.88e3T^{2} \)
89 \( 1 + 75.0T + 7.92e3T^{2} \)
97 \( 1 + 162.T + 9.40e3T^{2} \)
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

−10.11972137322508105498296487227, −9.586323327434842063281708296538, −8.025204556876279200374712655641, −7.40885894399198069455415576025, −6.98724280271916894208106873102, −5.77351452222984849539939320624, −4.73911836975744738867543666539, −4.15598856772392684449435119292, −3.03968488204979578917817062730, −1.63198134108980306597595108688, 0.60884530781620071184431401574, 2.44152210845263512338200475081, 2.86206316615390629165482248896, 4.25909326355010076577420282943, 5.28602284404974135355538996297, 5.86764527198734016909905943453, 6.75723372541261664347130745607, 7.990990422181684610725717372746, 8.984178350632974127564216286915, 9.670752378932781547603073221716

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