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Properties

Label 1-6048-6048.965-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  + (−0.906 − 0.422i)5-s + (−0.422 − 0.906i)11-s + (−0.996 + 0.0871i)13-s + (0.5 − 0.866i)17-s + (0.965 + 0.258i)19-s + (0.984 + 0.173i)23-s + (0.642 + 0.766i)25-s + (0.0871 − 0.996i)29-s + (−0.173 + 0.984i)31-s + (−0.965 + 0.258i)37-s + (0.642 − 0.766i)41-s + (0.422 + 0.906i)43-s + (−0.173 − 0.984i)47-s + (0.707 + 0.707i)53-s − i·55-s + ⋯
L(s)  = 1  + (−0.906 − 0.422i)5-s + (−0.422 − 0.906i)11-s + (−0.996 + 0.0871i)13-s + (0.5 − 0.866i)17-s + (0.965 + 0.258i)19-s + (0.984 + 0.173i)23-s + (0.642 + 0.766i)25-s + (0.0871 − 0.996i)29-s + (−0.173 + 0.984i)31-s + (−0.965 + 0.258i)37-s + (0.642 − 0.766i)41-s + (0.422 + 0.906i)43-s + (−0.173 − 0.984i)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: $\chi_{6048} (965, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ 0.468 - 0.883i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.078752601 - 0.6492481411i\)
\(L(\frac12)\) \(\approx\) \(1.078752601 - 0.6492481411i\)
\(L(1)\) \(\approx\) \(0.8754671990 - 0.1598208504i\)
\(L(1)\) \(\approx\) \(0.8754671990 - 0.1598208504i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-0.906 - 0.422i)T \)
11 \( 1 + (-0.422 - 0.906i)T \)
13 \( 1 + (-0.996 + 0.0871i)T \)
17 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (0.965 + 0.258i)T \)
23 \( 1 + (0.984 + 0.173i)T \)
29 \( 1 + (0.0871 - 0.996i)T \)
31 \( 1 + (-0.173 + 0.984i)T \)
37 \( 1 + (-0.965 + 0.258i)T \)
41 \( 1 + (0.642 - 0.766i)T \)
43 \( 1 + (0.422 + 0.906i)T \)
47 \( 1 + (-0.173 - 0.984i)T \)
53 \( 1 + (0.707 + 0.707i)T \)
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 \( 1 + (0.0871 - 0.996i)T \)
89 \( 1 + (0.866 - 0.5i)T \)
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

−17.87428034495174312995090600090, −17.14279021558009239035731207164, −16.51620667839681729296690481572, −15.711103770135157552303625464732, −15.17556435058195948611297672437, −14.679832875556331101265192990573, −14.09051486619218606008115476469, −13.0095924121771962917505547456, −12.49151524181992675058681144359, −12.001897168076355223060804441379, −11.164355111803209597591340990656, −10.593121711852929938467165616380, −9.90127434138841529879591183535, −9.224790727552642544698371883564, −8.343509343402975331577987043082, −7.52638036679329419427433514818, −7.31379497100276710979842410227, −6.52286413268030287980497910418, −5.447609263468288586937843561394, −4.87635437630093601580787906000, −4.13914117555513830542223859898, −3.30738407370104905478127887105, −2.6983597863012888751989319776, −1.80835764990487516911166412040, −0.69903299603827436724906862872, 0.504834672989390235975828378416, 1.187040322261309850305699211107, 2.467658345529966956378846570158, 3.1728932437564742574668873406, 3.74294830127776783593852820891, 4.861661601103545381391080265214, 5.12596192798363067759591767340, 5.9792989846057660454124385417, 7.15781912131273190184022173017, 7.42311160014448470524489315996, 8.18899783145200771236293820230, 8.907170303623157042116156501793, 9.515752970902356781020285698948, 10.34963229735169614184243089621, 11.069626479152591394562622863993, 11.841141636783015608449291106157, 12.09975286734644652659078594796, 12.97148105516361804481765543926, 13.67711310913264210936471683757, 14.31509620653031096409725266566, 15.02010935536429297243227562697, 15.88767029616907101174764214553, 16.071997663563248262566269702856, 16.90850337596102369331632847110, 17.42600337974245615741945907837

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