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Properties

Label 1-6048-6048.5981-r0-0-0
Degree $1$
Conductor $6048$
Sign $0.162 - 0.986i$
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  + (0.906 − 0.422i)5-s + (0.422 − 0.906i)11-s + (−0.422 − 0.906i)13-s + (0.5 − 0.866i)17-s + (0.258 − 0.965i)19-s + (0.984 − 0.173i)23-s + (0.642 − 0.766i)25-s + (0.906 + 0.422i)29-s + (0.939 + 0.342i)31-s + (−0.707 + 0.707i)37-s + (0.342 − 0.939i)41-s + (−0.573 − 0.819i)43-s + (0.939 − 0.342i)47-s + (0.965 + 0.258i)53-s − i·55-s + ⋯
L(s)  = 1  + (0.906 − 0.422i)5-s + (0.422 − 0.906i)11-s + (−0.422 − 0.906i)13-s + (0.5 − 0.866i)17-s + (0.258 − 0.965i)19-s + (0.984 − 0.173i)23-s + (0.642 − 0.766i)25-s + (0.906 + 0.422i)29-s + (0.939 + 0.342i)31-s + (−0.707 + 0.707i)37-s + (0.342 − 0.939i)41-s + (−0.573 − 0.819i)43-s + (0.939 − 0.342i)47-s + (0.965 + 0.258i)53-s − i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.162 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.162 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $0.162 - 0.986i$
Analytic conductor: \(28.0867\)
Root analytic conductor: \(28.0867\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6048} (5981, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ 0.162 - 0.986i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.002301476 - 1.699710101i\)
\(L(\frac12)\) \(\approx\) \(2.002301476 - 1.699710101i\)
\(L(1)\) \(\approx\) \(1.350461153 - 0.4115345782i\)
\(L(1)\) \(\approx\) \(1.350461153 - 0.4115345782i\)

Euler product

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

−17.74272912706753761133136877158, −17.13973753292906372354152500542, −16.83386446016356787672974948737, −15.881754400252539050687517662099, −15.0534085866351229620982410675, −14.481897816275358617491558037669, −14.11569539552166226455637673469, −13.27175694135471034232416773943, −12.58024910389166722680749056835, −11.98926801955200677495736554432, −11.253074601778004799610459933433, −10.38450059855858971604307399353, −9.85877205525338917445865136567, −9.42309577160075448701761566955, −8.56101164061464824268043825276, −7.733397170002242898285413088603, −6.94402755515375492357740511292, −6.44006354149877243483688257416, −5.74633557419226060423009164942, −4.91472700184623520298014429193, −4.22294405795843493911592640680, −3.36337760031696067619778342895, −2.48947417445371586289219380634, −1.79476434509464778603744014948, −1.15772686540596243748813238975, 0.791319566696919404873742956299, 1.07639690171085989776741186196, 2.486598074725819811112337510641, 2.836246154159914603108855295856, 3.74710033358394420026795134647, 4.99445410990370612515002650641, 5.13791168525303355460575640697, 5.97728400271198001510150850839, 6.79406001980482203662548555457, 7.342559834492457603985449692483, 8.50936778402921435142457706641, 8.78069553816958466781353006291, 9.55520053830980957946542389875, 10.31263310007956210391207027640, 10.74874164208670776809836978470, 11.80371142431505437022659486837, 12.217640075989957274298344534685, 13.08797762099791806256332453215, 13.77389075323443007784919962073, 13.96581813831902864367229770264, 14.989867503401806077123161022313, 15.59476845556761082333495362824, 16.383515383943269373517023126856, 16.94034341670330876022033892325, 17.52544235285238381328273480486

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