Properties

Label 1-6048-6048.1717-r0-0-0
Degree $1$
Conductor $6048$
Sign $0.879 - 0.476i$
Analytic cond. $28.0867$
Root an. cond. $28.0867$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.819 + 0.573i)5-s + (−0.573 + 0.819i)11-s + (0.573 + 0.819i)13-s + (0.5 + 0.866i)17-s + (−0.258 − 0.965i)19-s + (0.642 − 0.766i)23-s + (0.342 − 0.939i)25-s + (0.819 + 0.573i)29-s + (0.173 − 0.984i)31-s + (−0.707 − 0.707i)37-s + (−0.984 − 0.173i)41-s + (0.996 + 0.0871i)43-s + (−0.173 − 0.984i)47-s + (−0.965 + 0.258i)53-s i·55-s + ⋯
L(s)  = 1  + (−0.819 + 0.573i)5-s + (−0.573 + 0.819i)11-s + (0.573 + 0.819i)13-s + (0.5 + 0.866i)17-s + (−0.258 − 0.965i)19-s + (0.642 − 0.766i)23-s + (0.342 − 0.939i)25-s + (0.819 + 0.573i)29-s + (0.173 − 0.984i)31-s + (−0.707 − 0.707i)37-s + (−0.984 − 0.173i)41-s + (0.996 + 0.0871i)43-s + (−0.173 − 0.984i)47-s + (−0.965 + 0.258i)53-s i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.879 - 0.476i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.879 - 0.476i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $0.879 - 0.476i$
Analytic conductor: \(28.0867\)
Root analytic conductor: \(28.0867\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6048} (1717, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ 0.879 - 0.476i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.142371283 - 0.2896181619i\)
\(L(\frac12)\) \(\approx\) \(1.142371283 - 0.2896181619i\)
\(L(1)\) \(\approx\) \(0.8962532216 + 0.08512189142i\)
\(L(1)\) \(\approx\) \(0.8962532216 + 0.08512189142i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-0.819 + 0.573i)T \)
11 \( 1 + (-0.573 + 0.819i)T \)
13 \( 1 + (0.573 + 0.819i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (-0.258 - 0.965i)T \)
23 \( 1 + (0.642 - 0.766i)T \)
29 \( 1 + (0.819 + 0.573i)T \)
31 \( 1 + (0.173 - 0.984i)T \)
37 \( 1 + (-0.707 - 0.707i)T \)
41 \( 1 + (-0.984 - 0.173i)T \)
43 \( 1 + (0.996 + 0.0871i)T \)
47 \( 1 + (-0.173 - 0.984i)T \)
53 \( 1 + (-0.965 + 0.258i)T \)
59 \( 1 + (0.906 + 0.422i)T \)
61 \( 1 + (0.819 + 0.573i)T \)
67 \( 1 + (0.996 - 0.0871i)T \)
71 \( 1 + (-0.866 - 0.5i)T \)
73 \( 1 - iT \)
79 \( 1 + (-0.766 + 0.642i)T \)
83 \( 1 + (-0.819 - 0.573i)T \)
89 \( 1 + (-0.866 - 0.5i)T \)
97 \( 1 + (0.766 - 0.642i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−17.62447394846098757348258006961, −17.13467628046622787415146883287, −16.1856259379770199987602556879, −15.87596013570169247651912795514, −15.41816677866924213086302420983, −14.39758075528618219197787690960, −13.85476750643091264368681671878, −13.004323631024227888193027595374, −12.60723343667229037470857266866, −11.71874705285688344179051082173, −11.308808009156425737475529320, −10.47132744827451497765789677058, −9.8727184723042618223544263952, −8.910368653638721412545985944896, −8.26261956144730930784785512097, −7.93894191444469124644702531918, −7.10048316749307584018390230303, −6.22235437122738873418529932802, −5.32530304937643102861952119201, −5.02867866502788605143125441015, −3.929581039958510269676856289016, −3.316962821442636687254342240957, −2.75024866560136584998707028618, −1.35985628133871682437583200076, −0.80776493507781175669049588680, 0.398605367034587961916609352801, 1.59816156836354175727721690909, 2.445648888749822807916340431244, 3.14859245043840744089403940490, 4.05473178606509492364732421252, 4.49685723685070429693021731211, 5.3792983562093839339479179155, 6.35740794314478757839438360574, 6.957088631548819579726074635656, 7.440096340037736845041315609907, 8.41041433942316845944643639278, 8.75173975545244286807354068221, 9.8134610194707363226384981048, 10.478100008183046304507073151974, 11.00805277676795786801008830855, 11.67719104098773264707387834469, 12.40550910751674416452707671900, 12.93947246930939207364735068980, 13.774394232867947381394560219090, 14.547053533506388562661444237827, 15.03424496178593278651842402014, 15.65952367132387276859588588061, 16.206649345802102556534638924709, 17.02201778447630157509078478147, 17.68034636801192949490378641482

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