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Properties

Label 1-6048-6048.5645-r0-0-0
Degree $1$
Conductor $6048$
Sign $-0.735 - 0.677i$
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  + (0.996 − 0.0871i)5-s + (0.0871 − 0.996i)11-s + (0.906 + 0.422i)13-s − 17-s + (−0.707 − 0.707i)19-s + (0.342 + 0.939i)23-s + (0.984 − 0.173i)25-s + (−0.422 − 0.906i)29-s + (−0.173 + 0.984i)31-s + (−0.965 − 0.258i)37-s + (−0.342 − 0.939i)41-s + (−0.819 − 0.573i)43-s + (−0.173 − 0.984i)47-s + (−0.965 − 0.258i)53-s − i·55-s + ⋯
L(s)  = 1  + (0.996 − 0.0871i)5-s + (0.0871 − 0.996i)11-s + (0.906 + 0.422i)13-s − 17-s + (−0.707 − 0.707i)19-s + (0.342 + 0.939i)23-s + (0.984 − 0.173i)25-s + (−0.422 − 0.906i)29-s + (−0.173 + 0.984i)31-s + (−0.965 − 0.258i)37-s + (−0.342 − 0.939i)41-s + (−0.819 − 0.573i)43-s + (−0.173 − 0.984i)47-s + (−0.965 − 0.258i)53-s − i·55-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $-0.735 - 0.677i$
Analytic conductor: \(28.0867\)
Root analytic conductor: \(28.0867\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6048} (5645, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ -0.735 - 0.677i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3977769407 - 1.018172344i\)
\(L(\frac12)\) \(\approx\) \(0.3977769407 - 1.018172344i\)
\(L(1)\) \(\approx\) \(1.074535112 - 0.1899982846i\)
\(L(1)\) \(\approx\) \(1.074535112 - 0.1899982846i\)

Euler product

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

−17.96845539178710709288507897764, −17.32077213270096862762364130709, −16.78366169699620656427903302879, −16.026243953334637551385422592395, −15.1999008831191257042742443421, −14.711288521333194336168110853313, −14.05877177700372085579342844607, −13.17989158706219572930775580965, −12.89461934986490496741804050039, −12.20938622612561735800306673964, −11.05758859212592389922718073613, −10.77454326988449159619333049089, −9.931215311898929332941871872433, −9.4063897153385222167826365342, −8.63487933764826349641010519757, −8.04595156038371580109916858732, −6.99169251257364675454839356483, −6.4534554133215828678919342726, −5.92118416727699678478281975144, −4.96332396637619838907409632042, −4.44405099028505595372369257793, −3.46846226778629489545701981636, −2.63818617590667789983627351917, −1.83571227034659080145424593027, −1.32224684174915950050063964904, 0.23893278654759408623882978631, 1.464625519692841943141680624818, 1.952506837037960474351528505655, 2.93618541751187273320129254080, 3.64409707083753315500447953151, 4.511682148337176050162370130924, 5.35353639218420613192605110739, 5.92828734075675701669417761867, 6.621873994360566389752585360875, 7.13989593645028719488733416205, 8.39802318476676977029112138265, 8.82539046280217464424217002091, 9.27202119525390401627577437561, 10.25719866678760719927969760432, 10.858161634524173826357470911524, 11.35011027863955240961258649562, 12.19376565894860436572863560610, 13.15494257521878080764311324418, 13.585347554919556178179424397861, 13.859845063845632649995383557033, 14.85039224983361009629490119449, 15.57343444451875598187452934577, 16.11087582163840855313810745830, 17.030165793843367728783671046961, 17.29131257054707902349833843932

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