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Properties

Label 1-6048-6048.797-r0-0-0
Degree $1$
Conductor $6048$
Sign $0.883 - 0.468i$
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 + (−0.422 − 0.906i)13-s + (0.5 + 0.866i)17-s + (−0.965 + 0.258i)19-s + (−0.642 − 0.766i)23-s + (0.342 + 0.939i)25-s + (0.906 + 0.422i)29-s + (−0.766 + 0.642i)31-s + (0.965 + 0.258i)37-s + (0.342 − 0.939i)41-s + (−0.573 − 0.819i)43-s + (−0.766 − 0.642i)47-s + (−0.707 + 0.707i)53-s − i·55-s + ⋯
L(s)  = 1  + (−0.819 − 0.573i)5-s + (0.573 + 0.819i)11-s + (−0.422 − 0.906i)13-s + (0.5 + 0.866i)17-s + (−0.965 + 0.258i)19-s + (−0.642 − 0.766i)23-s + (0.342 + 0.939i)25-s + (0.906 + 0.422i)29-s + (−0.766 + 0.642i)31-s + (0.965 + 0.258i)37-s + (0.342 − 0.939i)41-s + (−0.573 − 0.819i)43-s + (−0.766 − 0.642i)47-s + (−0.707 + 0.707i)53-s − i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.883 - 0.468i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.883 - 0.468i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $0.883 - 0.468i$
Analytic conductor: \(28.0867\)
Root analytic conductor: \(28.0867\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6048} (797, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ 0.883 - 0.468i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.161225969 - 0.2886293509i\)
\(L(\frac12)\) \(\approx\) \(1.161225969 - 0.2886293509i\)
\(L(1)\) \(\approx\) \(0.8795450469 - 0.07152470559i\)
\(L(1)\) \(\approx\) \(0.8795450469 - 0.07152470559i\)

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 \( 1 + (-0.422 - 0.906i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (-0.965 + 0.258i)T \)
23 \( 1 + (-0.642 - 0.766i)T \)
29 \( 1 + (0.906 + 0.422i)T \)
31 \( 1 + (-0.766 + 0.642i)T \)
37 \( 1 + (0.965 + 0.258i)T \)
41 \( 1 + (0.342 - 0.939i)T \)
43 \( 1 + (-0.573 - 0.819i)T \)
47 \( 1 + (-0.766 - 0.642i)T \)
53 \( 1 + (-0.707 + 0.707i)T \)
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 \( 1 + (0.866 + 0.5i)T \)
79 \( 1 + (0.939 - 0.342i)T \)
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

−17.89322427945291188945132025581, −16.95218808962992074415714701042, −16.324074202928101506554823137087, −15.98983349862463440573116876263, −14.93101573264008114791359853672, −14.61632013680232141554246695375, −13.92107037007957563538157736059, −13.253316433779347234703858272174, −12.30979536828724347403300010835, −11.644368760964516069667396132277, −11.34767845693189831770892500529, −10.62573296042504034857615554286, −9.61363706513259374121486866611, −9.2674273531897341076325240981, −8.14483404536981559136847004872, −7.8627189948968042477200971264, −6.87802045801600640231094547744, −6.44521091257447541238958278313, −5.65375827261567201960061737509, −4.52451480733472914135105102428, −4.1650273996409053646485265502, −3.241378600110579973555028919970, −2.66602898054575175059252320320, −1.67069695361633668586249691697, −0.59615489966852643552290930685, 0.50105159340346739039534849121, 1.5137775317215812138752217790, 2.26441600242022631899434995010, 3.39757778426242122165455538219, 3.90797818409080691850647608120, 4.71662961367819184802853665408, 5.231288990117830316653149685404, 6.28575905551004249722421003273, 6.83094991475180727292901559032, 7.94849781549657929176053388275, 8.01955003819999993166432233065, 8.94584780563843716832971048375, 9.63679546840256129090614603915, 10.522977570876461467339837959505, 10.864370786316551261058728737697, 12.06779399859139812832392325470, 12.40191106035220320027342760654, 12.689855707462081148048880436208, 13.726757082503587794234304162072, 14.66450239159240472335014644015, 14.95809767968031717082277604420, 15.637199323533624414661810441255, 16.42376935663125798377631490972, 16.96855184847392181458555934896, 17.51862656858536328766507076455

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