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Properties

Label 1-6048-6048.277-r0-0-0
Degree $1$
Conductor $6048$
Sign $0.911 + 0.411i$
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 + (−0.996 + 0.0871i)13-s − 17-s + (−0.707 + 0.707i)19-s + (0.642 − 0.766i)23-s + (0.342 − 0.939i)25-s + (0.0871 − 0.996i)29-s + (−0.939 + 0.342i)31-s + (−0.258 + 0.965i)37-s + (0.642 − 0.766i)41-s + (−0.422 − 0.906i)43-s + (0.939 + 0.342i)47-s + (0.258 − 0.965i)53-s − i·55-s + ⋯
L(s)  = 1  + (−0.819 + 0.573i)5-s + (−0.573 + 0.819i)11-s + (−0.996 + 0.0871i)13-s − 17-s + (−0.707 + 0.707i)19-s + (0.642 − 0.766i)23-s + (0.342 − 0.939i)25-s + (0.0871 − 0.996i)29-s + (−0.939 + 0.342i)31-s + (−0.258 + 0.965i)37-s + (0.642 − 0.766i)41-s + (−0.422 − 0.906i)43-s + (0.939 + 0.342i)47-s + (0.258 − 0.965i)53-s − i·55-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $0.911 + 0.411i$
Analytic conductor: \(28.0867\)
Root analytic conductor: \(28.0867\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6048} (277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ 0.911 + 0.411i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6853142840 + 0.1474537905i\)
\(L(\frac12)\) \(\approx\) \(0.6853142840 + 0.1474537905i\)
\(L(1)\) \(\approx\) \(0.7010290560 + 0.09579216365i\)
\(L(1)\) \(\approx\) \(0.7010290560 + 0.09579216365i\)

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 \( 1 + (-0.996 + 0.0871i)T \)
17 \( 1 - T \)
19 \( 1 + (-0.707 + 0.707i)T \)
23 \( 1 + (0.642 - 0.766i)T \)
29 \( 1 + (0.0871 - 0.996i)T \)
31 \( 1 + (-0.939 + 0.342i)T \)
37 \( 1 + (-0.258 + 0.965i)T \)
41 \( 1 + (0.642 - 0.766i)T \)
43 \( 1 + (-0.422 - 0.906i)T \)
47 \( 1 + (0.939 + 0.342i)T \)
53 \( 1 + (0.258 - 0.965i)T \)
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 \( 1 + (-0.0871 + 0.996i)T \)
89 \( 1 + iT \)
97 \( 1 + (-0.939 - 0.342i)T \)
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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

−17.63092486914429631486469416390, −16.86486558238585518827903752453, −16.400663051564294058640407776761, −15.651223240140820895221939248729, −15.16775698734170180253337299832, −14.52958210409469782529638862917, −13.55751898797614293479025224592, −12.985803509539008326796601188929, −12.522289121410803408739924487741, −11.66574702832456756692217864484, −11.021450703055655342148542091230, −10.64735614250384880207115272274, −9.47475855831663103422243947996, −8.91837946369932684853262589845, −8.42550701234288458452724948374, −7.37264926984828176704874921090, −7.26400199215262707894779028458, −6.0839334403912681485627377336, −5.325265528693927333586805924592, −4.68229342856167408933071904596, −4.07003870985517129275420364932, −3.12341081372962081697020082602, −2.515154148600239726602626037750, −1.44279624612849254359594097601, −0.412027232221406097665075788, 0.39486815633996730431529528059, 1.941082438865783882897885471811, 2.4098585686396516286730401971, 3.25401908121543657239163928656, 4.20980272048864552696717997818, 4.59977546319341181659427098098, 5.46138678169553089651528548564, 6.51732745174155807867969319888, 6.99445222528291552010439384660, 7.64009071866176782727194554494, 8.28381686298281216383503770019, 9.056068400434894917815972595322, 9.901446537219524716485449550325, 10.58430305596088799428596458060, 10.982073837651713660107771318, 11.99305289467777329829770720696, 12.34588885158174346307385631945, 13.03900088156734881157416944028, 13.910613802823645305497103319881, 14.668455584783296100555246631110, 15.18333561632583534753664561681, 15.54367575303454614791101658045, 16.46537041232255658226516462186, 17.09176366733260216303593030912, 17.77900292289236509020476347841

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