Properties

Label 1-6048-6048.5387-r0-0-0
Degree $1$
Conductor $6048$
Sign $-0.987 + 0.157i$
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 + (0.906 − 0.422i)13-s + (−0.5 + 0.866i)17-s + (−0.965 − 0.258i)19-s + (0.984 − 0.173i)23-s + (−0.642 + 0.766i)25-s + (0.422 − 0.906i)29-s + (0.939 + 0.342i)31-s + (−0.707 − 0.707i)37-s + (0.342 − 0.939i)41-s + (−0.819 + 0.573i)43-s + (0.939 − 0.342i)47-s + (−0.258 + 0.965i)53-s + i·55-s + ⋯
L(s)  = 1  + (−0.422 − 0.906i)5-s + (−0.906 − 0.422i)11-s + (0.906 − 0.422i)13-s + (−0.5 + 0.866i)17-s + (−0.965 − 0.258i)19-s + (0.984 − 0.173i)23-s + (−0.642 + 0.766i)25-s + (0.422 − 0.906i)29-s + (0.939 + 0.342i)31-s + (−0.707 − 0.707i)37-s + (0.342 − 0.939i)41-s + (−0.819 + 0.573i)43-s + (0.939 − 0.342i)47-s + (−0.258 + 0.965i)53-s + i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.987 + 0.157i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.987 + 0.157i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $-0.987 + 0.157i$
Analytic conductor: \(28.0867\)
Root analytic conductor: \(28.0867\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6048} (5387, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ -0.987 + 0.157i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.04207714095 - 0.5318046579i\)
\(L(\frac12)\) \(\approx\) \(-0.04207714095 - 0.5318046579i\)
\(L(1)\) \(\approx\) \(0.8013704468 - 0.2369344400i\)
\(L(1)\) \(\approx\) \(0.8013704468 - 0.2369344400i\)

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 \( 1 + (0.906 - 0.422i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (-0.965 - 0.258i)T \)
23 \( 1 + (0.984 - 0.173i)T \)
29 \( 1 + (0.422 - 0.906i)T \)
31 \( 1 + (0.939 + 0.342i)T \)
37 \( 1 + (-0.707 - 0.707i)T \)
41 \( 1 + (0.342 - 0.939i)T \)
43 \( 1 + (-0.819 + 0.573i)T \)
47 \( 1 + (0.939 - 0.342i)T \)
53 \( 1 + (-0.258 + 0.965i)T \)
59 \( 1 + (-0.996 - 0.0871i)T \)
61 \( 1 + (-0.422 + 0.906i)T \)
67 \( 1 + (-0.819 - 0.573i)T \)
71 \( 1 + (0.866 - 0.5i)T \)
73 \( 1 - iT \)
79 \( 1 + (0.173 - 0.984i)T \)
83 \( 1 + (0.422 - 0.906i)T \)
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

−18.19948440038976066108698974635, −17.4911720298290878312096965723, −16.71433437007530392490528790917, −15.877105583287795435637419430760, −15.45524775608763025679296018288, −14.92410388287906485501948914877, −14.050427911214261059889645792498, −13.568063660518021111617967061731, −12.81236309056889736755976656652, −12.083279470543997662758352365393, −11.250182532825656465846826538047, −10.89743641316172698874581209913, −10.20699657307883872514329681, −9.48906339184326945547957521598, −8.53221845107939115485028542956, −8.10396495481241435883408016591, −7.10202791817693081051139229287, −6.77214017840748823970042618021, −6.020120547214208880758887327, −5.00942887162412432825894987825, −4.437848901774607156820628979597, −3.53601996098153325950128280026, −2.85391727529232803937777382232, −2.21777094719940555100732570412, −1.174610990862223265880658340904, 0.15205509596810842077932843907, 1.0174035558155787662063775761, 1.93710101859807695005316943543, 2.86177862792448042699899035121, 3.66543936675028289641486498917, 4.436802139290523208280135547, 4.98936637010049436088652799370, 5.90064542564416128205575099484, 6.3620614801351836897457528401, 7.48630735093328148256822561205, 8.05479798957388630252287497917, 8.80628361386450925026775244025, 8.9583100185339354495523104527, 10.316014076398648602386211359915, 10.67789412479095403333085664032, 11.34661883630289710687617345011, 12.28423308008966636513311218244, 12.72456943187975619708874974447, 13.44417321176738863158253499162, 13.77360546000362386186728852620, 15.111981725726732963485407483470, 15.38741775311855417630690799994, 15.961746318781618871268530173699, 16.74222462683737891454016506178, 17.28744902879439642545071587731

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