Properties

Label 1-6048-6048.5395-r0-0-0
Degree $1$
Conductor $6048$
Sign $0.995 - 0.0902i$
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  + (0.906 − 0.422i)5-s + (−0.422 + 0.906i)11-s + (0.573 − 0.819i)13-s + 17-s + (0.707 + 0.707i)19-s + (0.984 − 0.173i)23-s + (0.642 − 0.766i)25-s + (−0.819 + 0.573i)29-s + (0.766 − 0.642i)31-s + (0.258 + 0.965i)37-s + (0.984 − 0.173i)41-s + (0.996 − 0.0871i)43-s + (−0.766 − 0.642i)47-s + (−0.258 − 0.965i)53-s i·55-s + ⋯
L(s)  = 1  + (0.906 − 0.422i)5-s + (−0.422 + 0.906i)11-s + (0.573 − 0.819i)13-s + 17-s + (0.707 + 0.707i)19-s + (0.984 − 0.173i)23-s + (0.642 − 0.766i)25-s + (−0.819 + 0.573i)29-s + (0.766 − 0.642i)31-s + (0.258 + 0.965i)37-s + (0.984 − 0.173i)41-s + (0.996 − 0.0871i)43-s + (−0.766 − 0.642i)47-s + (−0.258 − 0.965i)53-s i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0902i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0902i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $0.995 - 0.0902i$
Analytic conductor: \(28.0867\)
Root analytic conductor: \(28.0867\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6048} (5395, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ 0.995 - 0.0902i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.642201911 - 0.1194948607i\)
\(L(\frac12)\) \(\approx\) \(2.642201911 - 0.1194948607i\)
\(L(1)\) \(\approx\) \(1.416422297 - 0.05533827221i\)
\(L(1)\) \(\approx\) \(1.416422297 - 0.05533827221i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (0.906 - 0.422i)T \)
11 \( 1 + (-0.422 + 0.906i)T \)
13 \( 1 + (0.573 - 0.819i)T \)
17 \( 1 + T \)
19 \( 1 + (0.707 + 0.707i)T \)
23 \( 1 + (0.984 - 0.173i)T \)
29 \( 1 + (-0.819 + 0.573i)T \)
31 \( 1 + (0.766 - 0.642i)T \)
37 \( 1 + (0.258 + 0.965i)T \)
41 \( 1 + (0.984 - 0.173i)T \)
43 \( 1 + (0.996 - 0.0871i)T \)
47 \( 1 + (-0.766 - 0.642i)T \)
53 \( 1 + (-0.258 - 0.965i)T \)
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 \( 1 + (0.819 - 0.573i)T \)
89 \( 1 - iT \)
97 \( 1 + (-0.766 - 0.642i)T \)
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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

−17.64584171419156703221021482967, −17.1885164781771637857093268540, −16.26302257191166293154247594870, −15.98861789688037055057367896203, −14.99199880264091853855817927310, −14.24164301931761185799336112718, −13.89722145050357011677490526277, −13.19960694560998621496494585611, −12.634353920042236223650112204213, −11.54675098284364094350061568137, −11.10165397534215657547588999937, −10.533379361791986716749582030717, −9.536204460445370791769990723, −9.29065128865618884533528176902, −8.416997665323758208448300024746, −7.56448724092686259792879209993, −6.93200544272686058452269088942, −6.08147727539971781107926967747, −5.65632389827337423427372863570, −4.9133936928310044606193671308, −3.92253552790078105364792775608, −3.05159017726523336827464853215, −2.615269136648296655498718006914, −1.52998943241300154544019036157, −0.8580408899202381644197403800, 0.89944153854752048990129517685, 1.4605485750034990057204897438, 2.43432877485902020223067396005, 3.10621106213593193223380804258, 4.00343070799628574541114995558, 4.97509769079163839971130919069, 5.44388593967538905464586124928, 6.03741939661195132089403123043, 6.93073529865840548811041906018, 7.723228458649997205967561988038, 8.26727690046294463306368628329, 9.181274373932172317246936240598, 9.77147167932168216102593379919, 10.26895728712457180834838603355, 10.9435964226094533007028431431, 11.93131510062322155193775781071, 12.51778092658090354675660855864, 13.175944327338963718450934376411, 13.54496257649166744466729064697, 14.62854053432242183736405617530, 14.838871945709176737092928505581, 15.8737100132897620224270487461, 16.3881667846787904210930980974, 17.13084208620321230947627363642, 17.640353564748686923683494420104

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