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Properties

Label 1-6048-6048.3125-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  + (−0.819 + 0.573i)5-s + (0.573 − 0.819i)11-s + (0.996 − 0.0871i)13-s − 17-s + (0.707 − 0.707i)19-s + (−0.642 + 0.766i)23-s + (0.342 − 0.939i)25-s + (−0.0871 + 0.996i)29-s + (0.939 − 0.342i)31-s + (−0.258 + 0.965i)37-s + (0.642 − 0.766i)41-s + (−0.422 − 0.906i)43-s + (0.939 + 0.342i)47-s + (−0.258 + 0.965i)53-s − i·55-s + ⋯
L(s)  = 1  + (−0.819 + 0.573i)5-s + (0.573 − 0.819i)11-s + (0.996 − 0.0871i)13-s − 17-s + (0.707 − 0.707i)19-s + (−0.642 + 0.766i)23-s + (0.342 − 0.939i)25-s + (−0.0871 + 0.996i)29-s + (0.939 − 0.342i)31-s + (−0.258 + 0.965i)37-s + (0.642 − 0.766i)41-s + (−0.422 − 0.906i)43-s + (0.939 + 0.342i)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: $\chi_{6048} (3125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ 0.907 - 0.419i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.469654600 - 0.3234078238i\)
\(L(\frac12)\) \(\approx\) \(1.469654600 - 0.3234078238i\)
\(L(1)\) \(\approx\) \(0.9908381575 + 0.002522576578i\)
\(L(1)\) \(\approx\) \(0.9908381575 + 0.002522576578i\)

Euler product

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

−17.77847709142667139506086359793, −17.08153305644438838679893207690, −16.20012888006342324824401518508, −15.96229605048925150056945111056, −15.20855155389599550746734497671, −14.54588521543650272804454796936, −13.79830503868286230928036672407, −13.05678984856129935899729338912, −12.482507970563904592392496073681, −11.69746234754803640212296624963, −11.42922676427678735592750722858, −10.44013285681063421121644623282, −9.75705610113696768757141578396, −8.96123541438788491540842852893, −8.42604196447861009517914536449, −7.77325307770551476657005228177, −7.015430635345587697730820929367, −6.295525849570753957262699540665, −5.569833088892966843072904677800, −4.45073026621480991594417627999, −4.261994332494882674303990778938, −3.47121587944734887685455861167, −2.43538994157657812184896045731, −1.54920874737436808675043911484, −0.76370735787058942317718164226, 0.54025463081896742941954417129, 1.42952014031406702764179598484, 2.52973875553598130477271245376, 3.321370516789604487365248659212, 3.801784917705016367774230701016, 4.56193724227737511946641653332, 5.51005529038506152963643617447, 6.320654890610369503541595640917, 6.82120927662614054775705353360, 7.5919894189843890401150623407, 8.35218668889063631044573158314, 8.87213792425718972880848153996, 9.61099147490955741500065919154, 10.71679791011960494985206699438, 10.97000252878652375867373996720, 11.70389864781627193156106357533, 12.15523119118744075858005080904, 13.24688467321028005488725155827, 13.78314583096937235548833633419, 14.25559483710789772091217661740, 15.28358671876072663906931182492, 15.68717649823094110869564101279, 16.09857234323288532638613179557, 17.04896125692686402478294815505, 17.70788815607793138478034722912

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