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Properties

Label 1-6048-6048.979-r0-0-0
Degree $1$
Conductor $6048$
Sign $-0.0943 + 0.995i$
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  + (0.422 − 0.906i)5-s + (−0.906 + 0.422i)11-s + (−0.0871 − 0.996i)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 + (−0.996 − 0.0871i)29-s + (0.173 − 0.984i)31-s + (0.258 + 0.965i)37-s + (0.642 − 0.766i)41-s + (−0.906 + 0.422i)43-s + (−0.173 − 0.984i)47-s + (−0.707 + 0.707i)53-s − i·55-s + ⋯
L(s)  = 1  + (0.422 − 0.906i)5-s + (−0.906 + 0.422i)11-s + (−0.0871 − 0.996i)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 + (−0.996 − 0.0871i)29-s + (0.173 − 0.984i)31-s + (0.258 + 0.965i)37-s + (0.642 − 0.766i)41-s + (−0.906 + 0.422i)43-s + (−0.173 − 0.984i)47-s + (−0.707 + 0.707i)53-s − i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0943 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0943 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $-0.0943 + 0.995i$
Analytic conductor: \(28.0867\)
Root analytic conductor: \(28.0867\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6048} (979, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ -0.0943 + 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2592559657 + 0.2850018484i\)
\(L(\frac12)\) \(\approx\) \(0.2592559657 + 0.2850018484i\)
\(L(1)\) \(\approx\) \(0.8569642630 - 0.1597478374i\)
\(L(1)\) \(\approx\) \(0.8569642630 - 0.1597478374i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (0.422 - 0.906i)T \)
11 \( 1 + (-0.906 + 0.422i)T \)
13 \( 1 + (-0.0871 - 0.996i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (0.258 - 0.965i)T \)
23 \( 1 + (-0.984 - 0.173i)T \)
29 \( 1 + (-0.996 - 0.0871i)T \)
31 \( 1 + (0.173 - 0.984i)T \)
37 \( 1 + (0.258 + 0.965i)T \)
41 \( 1 + (0.642 - 0.766i)T \)
43 \( 1 + (-0.906 + 0.422i)T \)
47 \( 1 + (-0.173 - 0.984i)T \)
53 \( 1 + (-0.707 + 0.707i)T \)
59 \( 1 + (-0.422 + 0.906i)T \)
61 \( 1 + (-0.573 + 0.819i)T \)
67 \( 1 + (0.0871 + 0.996i)T \)
71 \( 1 + (-0.866 - 0.5i)T \)
73 \( 1 + (-0.866 + 0.5i)T \)
79 \( 1 + (0.766 - 0.642i)T \)
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

−17.81394513974291293044720460286, −16.78528672256787891938483821240, −16.153967903201553976512901563960, −15.706665487122407594472029933063, −14.75825979341217124115242919505, −14.171727334604956766338509905625, −13.79476337384660920010679297075, −13.034322845847464856886046933831, −12.2301110029985109113702162121, −11.41584394917285568990312031524, −10.98702642199492128675461389112, −10.23295431226654410993160005335, −9.59004852604767365425592647166, −9.03420975932858394708751542435, −7.92866170033874467630692218319, −7.54802246415484527270294990772, −6.642927938606801418730222012805, −6.12874446252018562013805876905, −5.36461641640844385265505193667, −4.60875291719016692337733689430, −3.61955185742688116896213375535, −3.05302025583525528074308770779, −2.14972408010766706128355069944, −1.64680135360596281113410518577, −0.10269115288849048152707251255, 0.878005392639871838721997651234, 1.91987232730058752502868968749, 2.46517914733069846580397869819, 3.43698834816413925024298083934, 4.43315141833722080724230765846, 4.867959202332731674333357314164, 5.77836528556897006436444474044, 6.10317412540318256367351007498, 7.31883402415864508076429718549, 7.862330886340655653030498637961, 8.53829664977182920730136662893, 9.18742734384739184648259431274, 10.03369405041551997156054321096, 10.38255208077667060282525064079, 11.29781259139988571775742293188, 12.073411148498875212519607502264, 12.75388053659241327286910506719, 13.335424719033497025677704811317, 13.56330206415597642746165253407, 14.84493524561418352187824104521, 15.24390803031464816646603702278, 15.903824855762962359004170302614, 16.60187010035031860905213455851, 17.30893418372815858856783185017, 17.78983295101999008255213634650

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