Properties

Label 1-6048-6048.2453-r0-0-0
Degree $1$
Conductor $6048$
Sign $-0.527 + 0.849i$
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 + (0.0871 − 0.996i)13-s + (0.5 − 0.866i)17-s + (−0.965 − 0.258i)19-s + (0.342 − 0.939i)23-s + (0.984 + 0.173i)25-s + (−0.996 + 0.0871i)29-s + (−0.766 + 0.642i)31-s + (−0.707 − 0.707i)37-s + (−0.642 − 0.766i)41-s + (−0.906 − 0.422i)43-s + (−0.766 − 0.642i)47-s + (−0.258 + 0.965i)53-s i·55-s + ⋯
L(s)  = 1  + (−0.996 − 0.0871i)5-s + (−0.0871 − 0.996i)11-s + (0.0871 − 0.996i)13-s + (0.5 − 0.866i)17-s + (−0.965 − 0.258i)19-s + (0.342 − 0.939i)23-s + (0.984 + 0.173i)25-s + (−0.996 + 0.0871i)29-s + (−0.766 + 0.642i)31-s + (−0.707 − 0.707i)37-s + (−0.642 − 0.766i)41-s + (−0.906 − 0.422i)43-s + (−0.766 − 0.642i)47-s + (−0.258 + 0.965i)53-s i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.527 + 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.527 + 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $-0.527 + 0.849i$
Analytic conductor: \(28.0867\)
Root analytic conductor: \(28.0867\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6048} (2453, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ -0.527 + 0.849i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1104331831 - 0.1985951861i\)
\(L(\frac12)\) \(\approx\) \(-0.1104331831 - 0.1985951861i\)
\(L(1)\) \(\approx\) \(0.6897290934 - 0.2115145535i\)
\(L(1)\) \(\approx\) \(0.6897290934 - 0.2115145535i\)

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 \( 1 + (0.0871 - 0.996i)T \)
17 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (-0.965 - 0.258i)T \)
23 \( 1 + (0.342 - 0.939i)T \)
29 \( 1 + (-0.996 + 0.0871i)T \)
31 \( 1 + (-0.766 + 0.642i)T \)
37 \( 1 + (-0.707 - 0.707i)T \)
41 \( 1 + (-0.642 - 0.766i)T \)
43 \( 1 + (-0.906 - 0.422i)T \)
47 \( 1 + (-0.766 - 0.642i)T \)
53 \( 1 + (-0.258 + 0.965i)T \)
59 \( 1 + (0.573 - 0.819i)T \)
61 \( 1 + (-0.996 + 0.0871i)T \)
67 \( 1 + (-0.906 + 0.422i)T \)
71 \( 1 + (-0.866 + 0.5i)T \)
73 \( 1 - iT \)
79 \( 1 + (0.939 - 0.342i)T \)
83 \( 1 + (-0.996 + 0.0871i)T \)
89 \( 1 + (0.866 - 0.5i)T \)
97 \( 1 + (0.939 - 0.342i)T \)
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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

−18.31094936074763555691578590549, −17.34556370430215027918562882611, −16.76397346705345573129737024902, −16.28225766038739060679630257041, −15.30616486087034983832741848006, −14.932301766165554241145462309582, −14.52502925324163621551415197068, −13.34775917723868871372439282036, −12.917504095123502808767744067716, −12.07740606675384337819269300528, −11.65111992564647977910701171379, −10.93683699046929057999233689172, −10.229094569759586555319939875129, −9.472523480273003387646752795881, −8.791318950205472164848493692854, −7.97125798117708944671484427951, −7.50545164077411476425546924784, −6.74258280394592705710350249109, −6.12341936292831558716380040179, −5.04249982966767745574319931467, −4.45589351340727120602212391362, −3.75152771379039294549509411960, −3.185214936837211311044847019200, −1.9108339271712796275352920443, −1.53101972673935213745070563514, 0.077643452820921988974315910444, 0.71297491452094784120943337473, 1.87184800859588377036157530687, 3.01237887267394701153426054770, 3.357491938725316800251531058947, 4.18750467191990504484273019216, 5.08877035371550784298244351906, 5.57451503772133581486121491055, 6.56923809422847381687401775973, 7.265065031837213291420479695288, 7.87322818751967630775458225694, 8.737122788739847950408145971522, 8.87669283029189835620408243544, 10.20290194105597113588326278517, 10.69140555947156901704357056008, 11.30541427789477280081925693123, 11.97157714063056765997437696627, 12.71744683214176007117685935452, 13.16126994905677457485315284879, 14.10134932165073387984541418110, 14.71476047177986536399027228033, 15.370543490048822327097099219890, 15.950528012801702719944691337133, 16.59445765478050874947310558358, 17.065484626965146480503833945165

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