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Properties

Label 1-6048-6048.5485-r0-0-0
Degree $1$
Conductor $6048$
Sign $-0.523 - 0.852i$
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  + (0.0871 − 0.996i)5-s + (−0.996 + 0.0871i)11-s + (0.996 + 0.0871i)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 + (−0.0871 − 0.996i)29-s + (0.766 − 0.642i)31-s + (0.707 − 0.707i)37-s + (0.642 + 0.766i)41-s + (0.422 − 0.906i)43-s + (−0.766 − 0.642i)47-s + (0.965 + 0.258i)53-s + i·55-s + ⋯
L(s)  = 1  + (0.0871 − 0.996i)5-s + (−0.996 + 0.0871i)11-s + (0.996 + 0.0871i)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 + (−0.0871 − 0.996i)29-s + (0.766 − 0.642i)31-s + (0.707 − 0.707i)37-s + (0.642 + 0.766i)41-s + (0.422 − 0.906i)43-s + (−0.766 − 0.642i)47-s + (0.965 + 0.258i)53-s + i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.523 - 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.523 - 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $-0.523 - 0.852i$
Analytic conductor: \(28.0867\)
Root analytic conductor: \(28.0867\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6048} (5485, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ -0.523 - 0.852i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8705361891 - 1.555976251i\)
\(L(\frac12)\) \(\approx\) \(0.8705361891 - 1.555976251i\)
\(L(1)\) \(\approx\) \(1.039445563 - 0.4136261554i\)
\(L(1)\) \(\approx\) \(1.039445563 - 0.4136261554i\)

Euler product

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

−18.05514762365473845006772728251, −17.45498272819696412327829412217, −16.4573304263067633350815747900, −15.978903212499441799921248967995, −15.24782975305354261721992158571, −14.69491120896021550980884441032, −14.01574943619903178213726258736, −13.373097490405581495922062351135, −12.770567396051069454823913754915, −11.93907195323781167113296536597, −11.16791209531025852314857715663, −10.61893167774819636117848393519, −10.1590322832195060047592220639, −9.39108230551986037009344031845, −8.39327028018372745820860391343, −7.886745708998457690206517020127, −7.24481798848644955854451650858, −6.33650670035688979436241914995, −5.85153732337897812317346851180, −5.16871775567353874678859115210, −4.08029432105854904497830737263, −3.3299608001257944356238155758, −2.92338224186592109990312708442, −1.85109394394911730752400645520, −1.132839905177043936473931816889, 0.531982001706314603294180448262, 1.04257498193268806347753504388, 2.30530082107725568038089418183, 2.75669227424470457333750181119, 3.9395489583971753046256199631, 4.50681653367513411700195678731, 5.30032808546772891345594318664, 5.769383974371223565994922360055, 6.69376047325226111520665255645, 7.51834901968315450695316816830, 8.22263663575076672322693724795, 8.71322998157922093847862406118, 9.547500734625830205948066750884, 10.01440626881351338404077910493, 11.0111221459346880667543047477, 11.49672141046115726469091456401, 12.295128538450971689144616409008, 13.00246572149922736207519374601, 13.4543801514958663419562753109, 13.977608641727603549289435686461, 15.06297146390287689047094634879, 15.58991355436586323031266707366, 16.31059451267794712080181556762, 16.57830721538168607301463999253, 17.59970187860230361995249644166

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