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Properties

Label 1-6048-6048.1301-r0-0-0
Degree $1$
Conductor $6048$
Sign $0.579 + 0.814i$
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 + (0.819 + 0.573i)13-s + (0.5 + 0.866i)17-s + (0.258 + 0.965i)19-s + (0.342 − 0.939i)23-s + (0.984 + 0.173i)25-s + (0.573 + 0.819i)29-s + (0.939 + 0.342i)31-s + (−0.258 + 0.965i)37-s + (0.984 − 0.173i)41-s + (0.0871 + 0.996i)43-s + (0.939 − 0.342i)47-s + (−0.707 − 0.707i)53-s − i·55-s + ⋯
L(s)  = 1  + (−0.996 − 0.0871i)5-s + (−0.0871 − 0.996i)11-s + (0.819 + 0.573i)13-s + (0.5 + 0.866i)17-s + (0.258 + 0.965i)19-s + (0.342 − 0.939i)23-s + (0.984 + 0.173i)25-s + (0.573 + 0.819i)29-s + (0.939 + 0.342i)31-s + (−0.258 + 0.965i)37-s + (0.984 − 0.173i)41-s + (0.0871 + 0.996i)43-s + (0.939 − 0.342i)47-s + (−0.707 − 0.707i)53-s − i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.579 + 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.579 + 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $0.579 + 0.814i$
Analytic conductor: \(28.0867\)
Root analytic conductor: \(28.0867\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6048} (1301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ 0.579 + 0.814i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.320951614 + 0.6815506803i\)
\(L(\frac12)\) \(\approx\) \(1.320951614 + 0.6815506803i\)
\(L(1)\) \(\approx\) \(0.9794428025 + 0.08399520901i\)
\(L(1)\) \(\approx\) \(0.9794428025 + 0.08399520901i\)

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 \( 1 + (0.819 + 0.573i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (0.258 + 0.965i)T \)
23 \( 1 + (0.342 - 0.939i)T \)
29 \( 1 + (0.573 + 0.819i)T \)
31 \( 1 + (0.939 + 0.342i)T \)
37 \( 1 + (-0.258 + 0.965i)T \)
41 \( 1 + (0.984 - 0.173i)T \)
43 \( 1 + (0.0871 + 0.996i)T \)
47 \( 1 + (0.939 - 0.342i)T \)
53 \( 1 + (-0.707 - 0.707i)T \)
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 \( 1 + (0.573 + 0.819i)T \)
89 \( 1 + (-0.866 - 0.5i)T \)
97 \( 1 + (-0.766 - 0.642i)T \)
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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

−17.63604511253509625851931416187, −17.05942544531476503480644958069, −15.89542163595707462912629551440, −15.759538198520305253179861179134, −15.20195505284423159369473801099, −14.38502937313228594839117696088, −13.61585883506817229511973725821, −13.03251002801722423752360559564, −12.12620267371737803674491656877, −11.826802951283259268584037783831, −10.98051781502499389063531489171, −10.46861453217738412995853057758, −9.512047293443498817090397763652, −9.01518246659439743098293523611, −8.07354628476654255228051073843, −7.498711228614710922969541075551, −7.08370969298293241986378229749, −6.10968188130041956032881091599, −5.29903489995237398640233462563, −4.53936995527533145427907456805, −3.973248309809598895780837023993, −3.04798219476852200715621156404, −2.53774972990324998364538349849, −1.27271002628210070707127902506, −0.505431616028983186795431592414, 0.90152920291560510527560535841, 1.4615615002281552128394268757, 2.81000167823066057062969123560, 3.38590416369636435510497704560, 4.05513048162788367849258873582, 4.72777248208205633012104168211, 5.68595864620452563185307768855, 6.351170119671305712211611509798, 6.98436347665923091146936001370, 8.07215320631400339032946263314, 8.29109304405830465518867957374, 8.90636449777789797587061512096, 9.92642446287261431759105712985, 10.716066515485081651976001264275, 11.12828613838860999955749085805, 11.90967951809006265359051259502, 12.46807635768644366902054997821, 13.10284969815133828508184839587, 14.135817870331679497858637801337, 14.34315940002334142093650304892, 15.31008449587848338223726698007, 15.93224333944851088145748683329, 16.441440618367768312330313062975, 16.89759583076057529013227233400, 17.88219802620057046068662241318

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