↑

Properties

Label 2-30-5.4-c11-0-5
Degree $2$
Conductor $30$
Sign $0.960 - 0.279i$
Analytic cond. $23.0502$
Root an. cond. $4.80107$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 32i·2-s + 243i·3-s − 1.02e3·4-s + (−6.70e3 + 1.95e3i)5-s − 7.77e3·6-s − 5.29e4i·7-s − 3.27e4i·8-s − 5.90e4·9-s + (−6.24e4 − 2.14e5i)10-s + 5.31e5·11-s − 2.48e5i·12-s + 3.73e5i·13-s + 1.69e6·14-s + (−4.74e5 − 1.63e6i)15-s + 1.04e6·16-s − 4.81e6i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + 0.577i·3-s − 0.5·4-s + (−0.960 + 0.279i)5-s − 0.408·6-s − 1.18i·7-s − 0.353i·8-s − 0.333·9-s + (−0.197 − 0.678i)10-s + 0.994·11-s − 0.288i·12-s + 0.279i·13-s + 0.841·14-s + (−0.161 − 0.554i)15-s + 0.250·16-s − 0.822i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 30 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 - 0.279i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.960 - 0.279i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(30\)    =    \(2 \cdot 3 \cdot 5\)
Sign: $0.960 - 0.279i$
Analytic conductor: \(23.0502\)
Root analytic conductor: \(4.80107\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{30} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 30,\ (\ :11/2),\ 0.960 - 0.279i)\)

Particular Values

\(L(6)\) \(\approx\) \(1.30294 + 0.185589i\)
\(L(\frac12)\) \(\approx\) \(1.30294 + 0.185589i\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 32iT \)
3 \( 1 - 243iT \)
5 \( 1 + (6.70e3 - 1.95e3i)T \)
good7 \( 1 + 5.29e4iT - 1.97e9T^{2} \)
11 \( 1 - 5.31e5T + 2.85e11T^{2} \)
13 \( 1 - 3.73e5iT - 1.79e12T^{2} \)
17 \( 1 + 4.81e6iT - 3.42e13T^{2} \)
19 \( 1 + 5.80e6T + 1.16e14T^{2} \)
23 \( 1 - 3.32e7iT - 9.52e14T^{2} \)
29 \( 1 - 1.30e8T + 1.22e16T^{2} \)
31 \( 1 - 6.18e7T + 2.54e16T^{2} \)
37 \( 1 + 1.37e8iT - 1.77e17T^{2} \)
41 \( 1 - 1.46e9T + 5.50e17T^{2} \)
43 \( 1 + 5.30e8iT - 9.29e17T^{2} \)
47 \( 1 + 1.34e9iT - 2.47e18T^{2} \)
53 \( 1 + 4.74e9iT - 9.26e18T^{2} \)
59 \( 1 + 4.37e9T + 3.01e19T^{2} \)
61 \( 1 + 7.78e9T + 4.35e19T^{2} \)
67 \( 1 + 4.98e9iT - 1.22e20T^{2} \)
71 \( 1 - 9.09e9T + 2.31e20T^{2} \)
73 \( 1 + 1.88e10iT - 3.13e20T^{2} \)
79 \( 1 - 4.50e10T + 7.47e20T^{2} \)
83 \( 1 + 6.88e10iT - 1.28e21T^{2} \)
89 \( 1 - 3.49e9T + 2.77e21T^{2} \)
97 \( 1 - 1.62e11iT - 7.15e21T^{2} \)
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

−14.64118014174156891973781706783, −13.74105188315851369517947148305, −11.90126640706818969726522986932, −10.66625531207514429476817855963, −9.235492466074933419862466630651, −7.73637711821863782111313745355, −6.63866991311177738180284371027, −4.57492410224302979885262832582, −3.62032084465942930265623517185, −0.59373097540416759429421709920, 1.04982967352556303803832571432, 2.69245310748741044106504448458, 4.36026020743116458999998859313, 6.21851511006189160768443984141, 8.130153303584963542772162270021, 9.047396311982597577665507292678, 10.95477871101098433487748564693, 12.17789193211557345450637017536, 12.58648048013555150497373938663, 14.36429642279859224226953976911

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