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Properties

Label 1-6048-6048.3883-r0-0-0
Degree $1$
Conductor $6048$
Sign $0.297 - 0.954i$
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  + (−0.422 − 0.906i)5-s + (0.906 + 0.422i)11-s + (0.819 + 0.573i)13-s + 17-s + (−0.707 + 0.707i)19-s + (−0.984 + 0.173i)23-s + (−0.642 + 0.766i)25-s + (−0.573 − 0.819i)29-s + (0.766 − 0.642i)31-s + (0.965 − 0.258i)37-s + (−0.984 + 0.173i)41-s + (−0.0871 − 0.996i)43-s + (−0.766 − 0.642i)47-s + (−0.965 + 0.258i)53-s − i·55-s + ⋯
L(s)  = 1  + (−0.422 − 0.906i)5-s + (0.906 + 0.422i)11-s + (0.819 + 0.573i)13-s + 17-s + (−0.707 + 0.707i)19-s + (−0.984 + 0.173i)23-s + (−0.642 + 0.766i)25-s + (−0.573 − 0.819i)29-s + (0.766 − 0.642i)31-s + (0.965 − 0.258i)37-s + (−0.984 + 0.173i)41-s + (−0.0871 − 0.996i)43-s + (−0.766 − 0.642i)47-s + (−0.965 + 0.258i)53-s − i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.297 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.297 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $0.297 - 0.954i$
Analytic conductor: \(28.0867\)
Root analytic conductor: \(28.0867\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6048} (3883, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ 0.297 - 0.954i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.250385479 - 0.9198261067i\)
\(L(\frac12)\) \(\approx\) \(1.250385479 - 0.9198261067i\)
\(L(1)\) \(\approx\) \(1.028886420 - 0.1831737449i\)
\(L(1)\) \(\approx\) \(1.028886420 - 0.1831737449i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-0.422 - 0.906i)T \)
11 \( 1 + (0.906 + 0.422i)T \)
13 \( 1 + (0.819 + 0.573i)T \)
17 \( 1 + T \)
19 \( 1 + (-0.707 + 0.707i)T \)
23 \( 1 + (-0.984 + 0.173i)T \)
29 \( 1 + (-0.573 - 0.819i)T \)
31 \( 1 + (0.766 - 0.642i)T \)
37 \( 1 + (0.965 - 0.258i)T \)
41 \( 1 + (-0.984 + 0.173i)T \)
43 \( 1 + (-0.0871 - 0.996i)T \)
47 \( 1 + (-0.766 - 0.642i)T \)
53 \( 1 + (-0.965 + 0.258i)T \)
59 \( 1 + (0.573 - 0.819i)T \)
61 \( 1 + (-0.996 + 0.0871i)T \)
67 \( 1 + (0.906 - 0.422i)T \)
71 \( 1 + (-0.866 + 0.5i)T \)
73 \( 1 + (0.866 - 0.5i)T \)
79 \( 1 + (-0.939 + 0.342i)T \)
83 \( 1 + (0.573 + 0.819i)T \)
89 \( 1 - iT \)
97 \( 1 + (-0.766 - 0.642i)T \)
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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

−17.86425713943615347180156190715, −17.28434654209633582850012809503, −16.33711071783820547629546951220, −15.99366463204155389128715177222, −15.049053286953293677780038510919, −14.65191965719469518236262664180, −14.01151641598059593399526288963, −13.32626880976450004013116439379, −12.50772284608713677931103515708, −11.788134939986441620925843178388, −11.23223840494773895869492105224, −10.62544263427172319989461076373, −9.98708440184701804060296445081, −9.1903591378021721657319039218, −8.29612389982670966971939861702, −7.93070774541798160713616544314, −6.96749702684209884910836168944, −6.355108128007696348646180973411, −5.89263986847656048622585690, −4.82170097103035113419470712890, −3.99367627314560021500610752126, −3.339422980684750258617383826074, −2.83935941462394908684898502142, −1.712176087679626919141359544707, −0.875600333555476341977370646533, 0.47197224738322257605329731274, 1.542440903959170156843369362315, 1.91733796520445213369159525253, 3.310778749008667531874009348921, 4.04579299044125051328181126338, 4.329410246488681538236054786424, 5.39013754041517657059279095546, 6.063408249266959163281589911812, 6.68742010720963554716473937008, 7.746995495963499462495236496786, 8.15892533958054662108746771453, 8.85604171828224961754101053236, 9.64592691155673845903806144449, 10.03934973358498537071655487273, 11.19181181981815522012151526363, 11.73625809603346954753415526618, 12.22220964929052349238673454484, 12.89847215243564512949198691701, 13.65477362486142546092521746003, 14.2594166047031804051052386846, 15.02768234635036853254483917568, 15.636100404529622234134641060280, 16.40277368323826053261359654908, 16.886002617541113490392796220417, 17.29182112438441063131398170328

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