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Properties

Label 1-6048-6048.5317-r0-0-0
Degree $1$
Conductor $6048$
Sign $0.411 - 0.911i$
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 + (−0.422 + 0.906i)13-s − 17-s + (0.707 − 0.707i)19-s + (0.342 + 0.939i)23-s + (−0.984 + 0.173i)25-s + (0.906 − 0.422i)29-s + (0.173 − 0.984i)31-s + (0.258 − 0.965i)37-s + (0.342 + 0.939i)41-s + (0.573 − 0.819i)43-s + (−0.173 − 0.984i)47-s + (−0.258 + 0.965i)53-s − i·55-s + ⋯
L(s)  = 1  + (0.0871 + 0.996i)5-s + (−0.996 − 0.0871i)11-s + (−0.422 + 0.906i)13-s − 17-s + (0.707 − 0.707i)19-s + (0.342 + 0.939i)23-s + (−0.984 + 0.173i)25-s + (0.906 − 0.422i)29-s + (0.173 − 0.984i)31-s + (0.258 − 0.965i)37-s + (0.342 + 0.939i)41-s + (0.573 − 0.819i)43-s + (−0.173 − 0.984i)47-s + (−0.258 + 0.965i)53-s − i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.411 - 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.411 - 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $0.411 - 0.911i$
Analytic conductor: \(28.0867\)
Root analytic conductor: \(28.0867\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6048} (5317, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ 0.411 - 0.911i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6380145210 - 0.4120748809i\)
\(L(\frac12)\) \(\approx\) \(0.6380145210 - 0.4120748809i\)
\(L(1)\) \(\approx\) \(0.8647318063 + 0.1247577865i\)
\(L(1)\) \(\approx\) \(0.8647318063 + 0.1247577865i\)

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 \( 1 + (-0.422 + 0.906i)T \)
17 \( 1 - T \)
19 \( 1 + (0.707 - 0.707i)T \)
23 \( 1 + (0.342 + 0.939i)T \)
29 \( 1 + (0.906 - 0.422i)T \)
31 \( 1 + (0.173 - 0.984i)T \)
37 \( 1 + (0.258 - 0.965i)T \)
41 \( 1 + (0.342 + 0.939i)T \)
43 \( 1 + (0.573 - 0.819i)T \)
47 \( 1 + (-0.173 - 0.984i)T \)
53 \( 1 + (-0.258 + 0.965i)T \)
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 \( 1 + (0.866 + 0.5i)T \)
79 \( 1 + (-0.766 + 0.642i)T \)
83 \( 1 + (-0.906 + 0.422i)T \)
89 \( 1 + iT \)
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

−17.77621952567628895745967700830, −17.27525549191673460672925502772, −16.33751107243883376733343692315, −15.971236142821326229312554276431, −15.36519288895984608408393066878, −14.55111902596016065327601168852, −13.775094072784733374336221753862, −13.14284238492154215735907461765, −12.53964223268706704231951379308, −12.17230604448949117618290650480, −11.17279431770120659365833446018, −10.41102092478124636867623214402, −9.97511473177449071813301065893, −9.04776938625426617135293397944, −8.48424515770689542184559882089, −7.86826309504329903436325670213, −7.20723813106140853546965763822, −6.18766985362887077241980819762, −5.57636388746186417649642066070, −4.684742073584644914891777981862, −4.56645449216759270434997651207, −3.179762013872007770423328450456, −2.68660141796228611357960797735, −1.65454713928561638641884029498, −0.84118581986789293074934827199, 0.21994605428983839362333439576, 1.59172463207528480212117802666, 2.54293951944265725196082780363, 2.79070060699259464589528376206, 3.88308834269655024659141007960, 4.57858131550462425098672317855, 5.398897839025928160564890218090, 6.16344225613785473895524525065, 6.86965344643245338060669758505, 7.44939466252228252459476683888, 8.03012433024307068484957089213, 9.14360456528369815205164739596, 9.53873897337935215806654025835, 10.3771198270937616728069332796, 11.04043300472142591112643336481, 11.471262151890122393651507370898, 12.22408193084369308325742916675, 13.22706916206490417830168063288, 13.66330551158153625378868074498, 14.20189445254518729491051704737, 15.159116924467552358745124164738, 15.48540770024457974671760830236, 16.12579852043525930324690364974, 17.07767880145683347157045100143, 17.65124819607949474393074310799

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