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Properties

Label 1-6048-6048.6005-r0-0-0
Degree $1$
Conductor $6048$
Sign $-0.847 + 0.531i$
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  + (−0.0871 − 0.996i)5-s + (−0.996 − 0.0871i)11-s + (−0.573 − 0.819i)13-s + (0.5 − 0.866i)17-s + (−0.965 − 0.258i)19-s + (−0.342 − 0.939i)23-s + (−0.984 + 0.173i)25-s + (−0.819 − 0.573i)29-s + (0.939 − 0.342i)31-s + (0.965 − 0.258i)37-s + (−0.984 − 0.173i)41-s + (0.996 + 0.0871i)43-s + (0.939 + 0.342i)47-s + (−0.707 − 0.707i)53-s − i·55-s + ⋯
L(s)  = 1  + (−0.0871 − 0.996i)5-s + (−0.996 − 0.0871i)11-s + (−0.573 − 0.819i)13-s + (0.5 − 0.866i)17-s + (−0.965 − 0.258i)19-s + (−0.342 − 0.939i)23-s + (−0.984 + 0.173i)25-s + (−0.819 − 0.573i)29-s + (0.939 − 0.342i)31-s + (0.965 − 0.258i)37-s + (−0.984 − 0.173i)41-s + (0.996 + 0.0871i)43-s + (0.939 + 0.342i)47-s + (−0.707 − 0.707i)53-s − i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.847 + 0.531i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.847 + 0.531i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $-0.847 + 0.531i$
Analytic conductor: \(28.0867\)
Root analytic conductor: \(28.0867\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6048} (6005, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ -0.847 + 0.531i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2081370286 - 0.7238728901i\)
\(L(\frac12)\) \(\approx\) \(-0.2081370286 - 0.7238728901i\)
\(L(1)\) \(\approx\) \(0.7557640685 - 0.3676723338i\)
\(L(1)\) \(\approx\) \(0.7557640685 - 0.3676723338i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-0.0871 - 0.996i)T \)
11 \( 1 + (-0.996 - 0.0871i)T \)
13 \( 1 + (-0.573 - 0.819i)T \)
17 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (-0.965 - 0.258i)T \)
23 \( 1 + (-0.342 - 0.939i)T \)
29 \( 1 + (-0.819 - 0.573i)T \)
31 \( 1 + (0.939 - 0.342i)T \)
37 \( 1 + (0.965 - 0.258i)T \)
41 \( 1 + (-0.984 - 0.173i)T \)
43 \( 1 + (0.996 + 0.0871i)T \)
47 \( 1 + (0.939 + 0.342i)T \)
53 \( 1 + (-0.707 - 0.707i)T \)
59 \( 1 + (-0.0871 - 0.996i)T \)
61 \( 1 + (0.906 - 0.422i)T \)
67 \( 1 + (-0.573 - 0.819i)T \)
71 \( 1 + (0.866 + 0.5i)T \)
73 \( 1 + (0.866 - 0.5i)T \)
79 \( 1 + (-0.173 - 0.984i)T \)
83 \( 1 + (-0.819 - 0.573i)T \)
89 \( 1 + (0.866 - 0.5i)T \)
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

−18.22820389116877949347639220745, −17.37574377423327052087401200094, −16.900599999047201826486519153893, −16.071768359626020002669186427131, −15.26308844105173714120328167844, −14.963577939832506496714751381867, −14.16994829769788366112710044178, −13.62992624841202132358477612004, −12.788121226203825981670142996237, −12.1815360609475673287283066081, −11.40175946926568869276725738789, −10.78038887479239278430798290318, −10.17343341448700457476361775249, −9.66459592231198223072148678833, −8.680973209192485579250916555796, −7.91769299710578326553683462002, −7.390284938423937111336778610034, −6.675728955084861464397346205727, −5.957579028900397887148993790694, −5.30575480841152767645456627160, −4.27728513367882741056455228641, −3.74159124395258048219864021053, −2.775989499291741491469359832018, −2.24927361900571154913190648455, −1.37427908573091680469460588463, 0.23410939277754566302079255460, 0.78832576427618384323431327307, 2.116159901419645846820038363085, 2.59317886817776806845682774700, 3.58756175979466546090580542218, 4.53522173473084177070640444747, 4.96890116457743268222266766786, 5.68621765990175340341478865396, 6.38645600159463349098762588603, 7.51246360054377188871839712151, 7.92951371764427326618498911958, 8.520086733348184742026518112804, 9.39087670099735360441705550421, 9.94557487131954782307721990480, 10.65129266533155655785735693482, 11.41778308798950520921868591009, 12.214192904566634378999859638247, 12.75522459341988363798569301067, 13.20137158168897783459768558047, 13.97060470531072105421806906591, 14.77702193785101602615317954522, 15.513725429379044705055516840786, 15.94812375210805703044692271348, 16.71951945402461750527777243854, 17.23748143258730095727837102704

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