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Properties

Label 1-6048-6048.3821-r0-0-0
Degree $1$
Conductor $6048$
Sign $0.468 + 0.883i$
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 + (0.422 + 0.906i)13-s + (0.5 + 0.866i)17-s + (0.965 − 0.258i)19-s + (−0.642 − 0.766i)23-s + (0.342 + 0.939i)25-s + (−0.906 − 0.422i)29-s + (−0.766 + 0.642i)31-s + (−0.965 − 0.258i)37-s + (0.342 − 0.939i)41-s + (0.573 + 0.819i)43-s + (−0.766 − 0.642i)47-s + (0.707 − 0.707i)53-s − i·55-s + ⋯
L(s)  = 1  + (0.819 + 0.573i)5-s + (−0.573 − 0.819i)11-s + (0.422 + 0.906i)13-s + (0.5 + 0.866i)17-s + (0.965 − 0.258i)19-s + (−0.642 − 0.766i)23-s + (0.342 + 0.939i)25-s + (−0.906 − 0.422i)29-s + (−0.766 + 0.642i)31-s + (−0.965 − 0.258i)37-s + (0.342 − 0.939i)41-s + (0.573 + 0.819i)43-s + (−0.766 − 0.642i)47-s + (0.707 − 0.707i)53-s − i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.468 + 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.468 + 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $0.468 + 0.883i$
Analytic conductor: \(28.0867\)
Root analytic conductor: \(28.0867\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6048} (3821, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ 0.468 + 0.883i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.742144782 + 1.048511272i\)
\(L(\frac12)\) \(\approx\) \(1.742144782 + 1.048511272i\)
\(L(1)\) \(\approx\) \(1.212336392 + 0.2108930956i\)
\(L(1)\) \(\approx\) \(1.212336392 + 0.2108930956i\)

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

−17.63578518488200710256423758031, −16.99311178702610215635179664446, −16.24590610539392079771841527186, −15.7164852939655555255755555411, −15.048164291340636914267097944008, −14.10729879982729073483595310664, −13.73306943334338378777564186339, −12.864993163322583565977874073810, −12.57021349320095029373311863473, −11.704134131704487044644177618982, −10.93587139449216810131825044251, −10.101003301533506684235057092718, −9.64141522519894532570544530449, −9.14138082451265761349156501683, −8.07019860013079337098118243524, −7.65946952561524949179750150197, −6.85761813886554362225536951933, −5.82554517274256567662215241600, −5.37870030484448735064272878641, −4.90445637080552693899369657825, −3.79817031773389743746136893113, −3.088794157523602549065049670690, −2.16572635724621980105101536735, −1.51473345674596965765484891006, −0.566872332684732113150146782102, 0.9216592610529852198296874902, 1.88399401382115939071797250087, 2.44261663266029611135063873305, 3.46486096495595848805021337914, 3.872089465810327949290368797894, 5.147176454026046821234571346810, 5.63429391507188834735032213782, 6.29017984453202971879045329541, 6.9835715681936850219343627164, 7.71366039869159692157039713706, 8.59288414309694131854091396026, 9.124534060465989760212810934012, 9.97423247432879553746363061017, 10.49749571912842162642390382480, 11.14343254041199011960889171517, 11.76803615742861464279499194518, 12.699486857800153437146188916893, 13.31401339754988507280566218832, 13.973972392977709121951999757528, 14.37798422676850160796413387137, 15.09721298403595206306965135783, 16.05871859112423801669523347285, 16.4191054792464470879357377325, 17.158569342116124426397506667171, 18.00820315909654785186047675187

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