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Properties

Label 1-6048-6048.4675-r0-0-0
Degree $1$
Conductor $6048$
Sign $0.883 - 0.468i$
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

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

Invariants

Degree: \(1\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $0.883 - 0.468i$
Analytic conductor: \(28.0867\)
Root analytic conductor: \(28.0867\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6048} (4675, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ 0.883 - 0.468i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.683604600 - 0.4184695451i\)
\(L(\frac12)\) \(\approx\) \(1.683604600 - 0.4184695451i\)
\(L(1)\) \(\approx\) \(1.055677153 - 0.05006555031i\)
\(L(1)\) \(\approx\) \(1.055677153 - 0.05006555031i\)

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

−17.82365219226097475910615089957, −16.90418317920574725506127192927, −16.5658755386956533704797347693, −15.65800333643335990686898917127, −15.17046206993302121308047341897, −14.77818132633598882957109594721, −13.62692600623801453530316898371, −13.16695750437095116608680239155, −12.48054253289303586994015582652, −11.74606793127486143173125882532, −11.30496445661375637533775007320, −10.57930957310364178467079971574, −9.62138708533351038841081196818, −9.15467281279708357558265345796, −8.24238199076847993556087455867, −7.85075594623210215936073452788, −7.033532789102987009106157788490, −6.321759536175697665173026576785, −5.530457108966176841754587545926, −4.62787776209780304604313225063, −4.08203335291591784832546169607, −3.48792700423395766644936636532, −2.515068807867527844403746160245, −1.480645005581678161633100037, −0.831835345596644738677843140100, 0.635557528258386499173425366495, 1.28810349046291985200175436972, 2.62522169686372054175169494332, 3.26192602273855122251396867629, 3.74951955272406407823285632170, 4.69456094071059805457624295420, 5.361025062376784095415406154536, 6.32289808601679898893232744917, 6.935058808425331667993156265802, 7.45856054635295910456138043116, 8.49144664360577276868212737434, 8.77639553631786454346349203725, 9.58002488750615500231918283393, 10.7072715787813823872180981537, 11.01223324682538392766710351993, 11.644333122631137995252973118041, 12.21760572878648114900158448937, 13.153041135547188171365642514934, 13.81368604645981853341674042929, 14.3005292565843090089312454192, 15.12326555334482880910246407164, 15.83269730210442719600312051369, 16.168641207991804720195988855884, 16.83079081622968422687217115687, 17.84701125860912618855616034682

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