↑

Properties

Label 1-6048-6048.1867-r0-0-0
Degree $1$
Conductor $6048$
Sign $0.677 + 0.735i$
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  + (−0.573 + 0.819i)5-s + (−0.819 + 0.573i)11-s + (0.0871 − 0.996i)13-s + 17-s + (−0.707 + 0.707i)19-s + (0.642 + 0.766i)23-s + (−0.342 − 0.939i)25-s + (0.996 − 0.0871i)29-s + (−0.939 − 0.342i)31-s + (0.965 − 0.258i)37-s + (0.642 + 0.766i)41-s + (0.906 + 0.422i)43-s + (0.939 − 0.342i)47-s + (−0.965 + 0.258i)53-s − i·55-s + ⋯
L(s)  = 1  + (−0.573 + 0.819i)5-s + (−0.819 + 0.573i)11-s + (0.0871 − 0.996i)13-s + 17-s + (−0.707 + 0.707i)19-s + (0.642 + 0.766i)23-s + (−0.342 − 0.939i)25-s + (0.996 − 0.0871i)29-s + (−0.939 − 0.342i)31-s + (0.965 − 0.258i)37-s + (0.642 + 0.766i)41-s + (0.906 + 0.422i)43-s + (0.939 − 0.342i)47-s + (−0.965 + 0.258i)53-s − i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.677 + 0.735i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.677 + 0.735i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $0.677 + 0.735i$
Analytic conductor: \(28.0867\)
Root analytic conductor: \(28.0867\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6048} (1867, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ 0.677 + 0.735i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.280315160 + 0.5609675777i\)
\(L(\frac12)\) \(\approx\) \(1.280315160 + 0.5609675777i\)
\(L(1)\) \(\approx\) \(0.9333975546 + 0.1640125581i\)
\(L(1)\) \(\approx\) \(0.9333975546 + 0.1640125581i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-0.573 + 0.819i)T \)
11 \( 1 + (-0.819 + 0.573i)T \)
13 \( 1 + (0.0871 - 0.996i)T \)
17 \( 1 + T \)
19 \( 1 + (-0.707 + 0.707i)T \)
23 \( 1 + (0.642 + 0.766i)T \)
29 \( 1 + (0.996 - 0.0871i)T \)
31 \( 1 + (-0.939 - 0.342i)T \)
37 \( 1 + (0.965 - 0.258i)T \)
41 \( 1 + (0.642 + 0.766i)T \)
43 \( 1 + (0.906 + 0.422i)T \)
47 \( 1 + (0.939 - 0.342i)T \)
53 \( 1 + (-0.965 + 0.258i)T \)
59 \( 1 + (-0.996 - 0.0871i)T \)
61 \( 1 + (0.422 - 0.906i)T \)
67 \( 1 + (-0.819 - 0.573i)T \)
71 \( 1 + (-0.866 + 0.5i)T \)
73 \( 1 + (0.866 - 0.5i)T \)
79 \( 1 + (0.173 - 0.984i)T \)
83 \( 1 + (-0.996 + 0.0871i)T \)
89 \( 1 - iT \)
97 \( 1 + (0.939 - 0.342i)T \)
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

−17.525800508776421853554299287959, −16.74176434686025252429663651301, −16.38645531135057161106272266339, −15.77631265325609527962472284805, −15.10176103373013070065542722899, −14.27430324418487454862724310770, −13.71793807925831892952029986956, −12.79691967991326493062259006840, −12.52475953686788504876113570639, −11.706565452794548908978754758457, −10.99098589497777533841483822022, −10.504340238141260433390145803353, −9.443093664018001483090430615549, −8.89022166419982742321218856428, −8.34652680872588999198619360938, −7.59615317084875641058413912288, −6.961214226237028668148794988539, −6.01614244114386704634409748095, −5.350279307264179075796182567955, −4.56299586934920129245127013654, −4.09255126439647089396213934345, −3.09357383553775336982146393521, −2.40434952430071151221997396558, −1.30212678380614720710217750199, −0.55988704642047674211708776893, 0.66487284759709543833229724888, 1.77634500515546961435114009410, 2.77310248013326793619397807365, 3.170342668388195502974781439858, 4.05165026332461841131992849131, 4.80444766154036607219874040800, 5.72038560733304726063161842763, 6.20447430282633833124382066794, 7.263104049467832150452071982794, 7.76288679680935243081482501877, 8.08993344589296838109421864260, 9.23529372682607302796083695861, 9.973333385323993752542122044765, 10.59282420589215916780367874968, 11.00183162511088630119692803186, 11.89414348068621994901105133465, 12.60487071096701471459356810216, 13.00413742034417708628437161469, 14.01002944984132126547292342405, 14.659218301419649556880108335115, 15.12795202054435787553261863360, 15.74519679663950126365137120797, 16.34831080154635186298446884484, 17.26657231372669908818521289922, 17.824294796705855155513009605429

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