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Properties

Label 1-6048-6048.2203-r0-0-0
Degree $1$
Conductor $6048$
Sign $-0.352 + 0.935i$
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 + (0.573 + 0.819i)13-s + (−0.5 − 0.866i)17-s + (0.258 + 0.965i)19-s + (−0.642 + 0.766i)23-s + (0.342 − 0.939i)25-s + (−0.819 − 0.573i)29-s + (0.173 − 0.984i)31-s + (0.707 + 0.707i)37-s + (0.984 + 0.173i)41-s + (0.996 + 0.0871i)43-s + (−0.173 − 0.984i)47-s + (0.965 − 0.258i)53-s − i·55-s + ⋯
L(s)  = 1  + (−0.819 + 0.573i)5-s + (−0.573 + 0.819i)11-s + (0.573 + 0.819i)13-s + (−0.5 − 0.866i)17-s + (0.258 + 0.965i)19-s + (−0.642 + 0.766i)23-s + (0.342 − 0.939i)25-s + (−0.819 − 0.573i)29-s + (0.173 − 0.984i)31-s + (0.707 + 0.707i)37-s + (0.984 + 0.173i)41-s + (0.996 + 0.0871i)43-s + (−0.173 − 0.984i)47-s + (0.965 − 0.258i)53-s − i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.352 + 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.352 + 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $-0.352 + 0.935i$
Analytic conductor: \(28.0867\)
Root analytic conductor: \(28.0867\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6048} (2203, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ -0.352 + 0.935i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6657984138 + 0.9625110533i\)
\(L(\frac12)\) \(\approx\) \(0.6657984138 + 0.9625110533i\)
\(L(1)\) \(\approx\) \(0.8532129724 + 0.2398461840i\)
\(L(1)\) \(\approx\) \(0.8532129724 + 0.2398461840i\)

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 \( 1 + (0.573 + 0.819i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (0.258 + 0.965i)T \)
23 \( 1 + (-0.642 + 0.766i)T \)
29 \( 1 + (-0.819 - 0.573i)T \)
31 \( 1 + (0.173 - 0.984i)T \)
37 \( 1 + (0.707 + 0.707i)T \)
41 \( 1 + (0.984 + 0.173i)T \)
43 \( 1 + (0.996 + 0.0871i)T \)
47 \( 1 + (-0.173 - 0.984i)T \)
53 \( 1 + (0.965 - 0.258i)T \)
59 \( 1 + (-0.906 - 0.422i)T \)
61 \( 1 + (0.819 + 0.573i)T \)
67 \( 1 + (0.996 - 0.0871i)T \)
71 \( 1 + (0.866 + 0.5i)T \)
73 \( 1 - iT \)
79 \( 1 + (0.766 - 0.642i)T \)
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

−17.57826726791802782004370846251, −16.69956145632012205370660212494, −16.0610108422378129973380093876, −15.705606777457718999322623479782, −15.04058489900882953445522232465, −14.22623950162343198017603174749, −13.44924718367948993593482773687, −12.754197748385108372790595692614, −12.490209618759625082062270273450, −11.426547966239859630668923715980, −10.85711719973959350527068422638, −10.5328377147912681397268977861, −9.23710122961491504843950806730, −8.83839903712778704569009367689, −8.04009204159512211497477435831, −7.70939769022834867534708820957, −6.70260015994754252537166856571, −5.89996143149425454891962776329, −5.28745584876601101916973218266, −4.466141916162569632318322262, −3.7847563591206303665278861621, −3.09139114473249364282886818009, −2.25175530770412416670477674098, −1.0714187032997680912308998530, −0.40762397037964547858765475118, 0.86139608333497618897627275149, 2.08417969027749885513522537124, 2.53559192106184592834844432699, 3.72841269775758446944004002105, 4.02096104178003517370208908605, 4.88259531785243472366129727051, 5.7863731322388561435462127052, 6.51235389174937634514630995438, 7.2884190540301695906710007401, 7.754311171772012692013272424, 8.38318336738130473910535494371, 9.485343641672624641169744171981, 9.79673372847010969665968622578, 10.76047994265573140896075109104, 11.4095762074182716935451787318, 11.8054641643980868315677897619, 12.567090534502492112546104974584, 13.41972601495483089940548021686, 13.94638342389396945884074154339, 14.78239681659773134079075977893, 15.26019882087087061693750090902, 15.982454440008398508587430363142, 16.3650866961151079538544730202, 17.29567916689979051809011924319, 18.11411162085646550191625569047

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