Properties

Label 1-6048-6048.619-r0-0-0
Degree $1$
Conductor $6048$
Sign $0.907 - 0.419i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.0871 + 0.996i)5-s + (−0.996 − 0.0871i)11-s + (−0.422 + 0.906i)13-s + 17-s + (−0.707 + 0.707i)19-s + (−0.342 − 0.939i)23-s + (−0.984 + 0.173i)25-s + (−0.906 + 0.422i)29-s + (0.173 − 0.984i)31-s + (−0.258 + 0.965i)37-s + (−0.342 − 0.939i)41-s + (0.573 − 0.819i)43-s + (−0.173 − 0.984i)47-s + (0.258 − 0.965i)53-s i·55-s + ⋯
L(s)  = 1  + (0.0871 + 0.996i)5-s + (−0.996 − 0.0871i)11-s + (−0.422 + 0.906i)13-s + 17-s + (−0.707 + 0.707i)19-s + (−0.342 − 0.939i)23-s + (−0.984 + 0.173i)25-s + (−0.906 + 0.422i)29-s + (0.173 − 0.984i)31-s + (−0.258 + 0.965i)37-s + (−0.342 − 0.939i)41-s + (0.573 − 0.819i)43-s + (−0.173 − 0.984i)47-s + (0.258 − 0.965i)53-s i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.907 - 0.419i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.907 - 0.419i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $0.907 - 0.419i$
Analytic conductor: \(28.0867\)
Root analytic conductor: \(28.0867\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6048} (619, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ 0.907 - 0.419i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.032286606 - 0.2271619228i\)
\(L(\frac12)\) \(\approx\) \(1.032286606 - 0.2271619228i\)
\(L(1)\) \(\approx\) \(0.8925849701 + 0.1355352447i\)
\(L(1)\) \(\approx\) \(0.8925849701 + 0.1355352447i\)

Euler product

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

Imaginary part of the first few zeros on the critical line

−17.736301336693934630973440544, −17.11949000172606179940234559507, −16.40998284424336230945489169513, −15.82777638269017443117823515618, −15.23355721367879787952222887895, −14.5546371165236411167551978677, −13.58890618238298467884894294094, −13.14607991816102504655084768446, −12.500631800547138740285399104527, −12.05956244950698927671565535647, −11.04472566217625180668118148070, −10.47720552384396044474144792687, −9.65122933801917008739652003513, −9.23289475620390057371697169694, −8.18591827220991682574324366293, −7.88195553662367956212382660757, −7.17497934819663204698785150944, −6.00110346810093118791738892249, −5.4802613985583495626249644369, −4.92817060772641094422453273196, −4.16703378425715445427257411008, −3.22145461342563109649318252587, −2.504457922476266829919313487987, −1.58081131703074318156216289993, −0.723269848752553613984306758201, 0.34145515604431020320637720592, 1.8973742507149801936473058444, 2.2285879700697397805381232320, 3.20934814992776877727936573151, 3.81647166061028011450377972798, 4.69351681549131332198732223157, 5.57756073198612427524231677637, 6.14361242949280459745434999279, 6.96389306407896029629109176987, 7.51359049946256433194930744573, 8.20723339391200600286117180906, 8.977621512613931400127466420329, 10.07007013105976443331517049124, 10.17851776745874417930914069600, 10.97021430026703258791740420247, 11.74897172901373477441734297191, 12.29847473621351000361787594328, 13.14026354684248736484618713111, 13.8212794917109410253744864137, 14.461793473569570157566609315244, 14.945312159746604594402425284804, 15.574079094731313883246754552972, 16.58300005063147141704161075422, 16.78486276465827418839291611315, 17.771729119115430139747436070940

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