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Properties

Label 1-6048-6048.2477-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: $\chi_{6048} (2477, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ 0.0943 - 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4170819656 - 0.3794045140i\)
\(L(\frac12)\) \(\approx\) \(0.4170819656 - 0.3794045140i\)
\(L(1)\) \(\approx\) \(0.8004666831 + 0.09266090483i\)
\(L(1)\) \(\approx\) \(0.8004666831 + 0.09266090483i\)

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

−17.69753906883036472080434249933, −17.18882537044392314595331787753, −16.46692322219506807616192423717, −15.89076565771089786054048906125, −15.496115175731925433462699713588, −14.53397624492070823260372838952, −13.8874081910558178300464336722, −13.16685271000616503812752402722, −12.646571422128476046890418169, −11.95002738709947161100811724839, −11.33921367866517812678517310053, −10.61686454417306838955206738125, −9.855187487061692877873922167747, −9.06031627765905460196160778428, −8.573630393974040288601049588544, −7.8017384234577713718569942982, −7.33749918845388382787954061844, −6.17494544768007151303695077373, −5.74523907346472815309629866611, −4.76033754372446567949012307431, −4.30700105282892080068250986983, −3.544015634325567574972706552810, −2.53464318880559072019566030266, −1.79404121387664829710334289494, −0.815033695850242581200285023606, 0.17413045500890258781124630429, 1.40514180479540162395712104792, 2.49627215669995367291134426142, 2.960575782161453070148076334719, 3.68940286267240112039873748995, 4.59879968633744921368211528210, 5.33859588810218116339138568215, 6.06336526827784875018217606233, 6.84095102974044643608085979611, 7.59442252487351172546015599776, 7.98767517741505249390914267229, 8.7154162172891193643864427670, 10.018093163020278273216017380568, 10.17787680918502210631986127737, 10.72470610271872721069586083507, 11.83258272189646391002114486115, 12.078935404185681954783637142250, 12.93572771226641900148988691614, 13.72063989691198277521618301985, 14.32216903906034530183892131117, 14.99494333763655504251120090604, 15.53528193420384756875962347320, 16.13600008965381844113620493825, 16.804341804364802084225476059135, 17.95626717797690206787550201374

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