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Properties

Label 1-6048-6048.373-r0-0-0
Degree $1$
Conductor $6048$
Sign $0.526 + 0.850i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.526 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.526 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $0.526 + 0.850i$
Analytic conductor: \(28.0867\)
Root analytic conductor: \(28.0867\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6048} (373, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ 0.526 + 0.850i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.128034685 + 0.6280425685i\)
\(L(\frac12)\) \(\approx\) \(1.128034685 + 0.6280425685i\)
\(L(1)\) \(\approx\) \(0.9521060036 + 0.1383056445i\)
\(L(1)\) \(\approx\) \(0.9521060036 + 0.1383056445i\)

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

−17.50211681485166534316955773409, −17.073421190088887128966259453017, −16.257131733131325731832589884985, −15.742783082437821486155618020986, −14.98344349319123593019594495409, −14.38240688644307192103979463187, −13.3947882637412896229453665912, −13.05174990578828834751439542563, −12.52393734578300146928718790288, −11.69901570982837996809358277098, −10.9564509652405465711160120543, −10.29343180122046836140427839959, −9.45047942754894488529093214453, −9.03978643770416939626891975268, −8.34808399181033234734086760567, −7.50333911902565629158528019578, −6.84002713799647108852996325638, −6.10687787888384672246239658558, −5.07710571260192303474695271084, −4.814914231582450877354076977552, −4.14232395627997278545463415579, −2.8671260168985801035031127445, −2.29036456261962594653641660561, −1.53429395612993446579915912876, −0.43961494152333395762267641520, 0.67661317023508426250795992804, 2.06669078560816423513863756453, 2.50768402947025178075661299917, 3.164113059361296918893755578875, 4.127470163066794440043718815075, 4.91503733042594921435254638488, 5.733737612529194186721773886029, 6.295162836802178134244159041683, 7.00806311667418887558118570422, 7.83892934337399031820385258888, 8.2336801677378493374056057100, 9.400513253703507370744216732577, 9.81055015599855555165863767700, 10.68570885429131730410987455311, 10.99348124325909167146073662589, 11.741685360785908199907044119, 12.881144044786327234654459531783, 13.073355639105799634739281433619, 13.86486994381178216598028159388, 14.77171566885880494231889549131, 14.98326905129939278589044839658, 15.749291642077628898670727637278, 16.551022551937967213091363917151, 17.30261978921924641105705008711, 17.789670912668483430670827394292

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