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Properties

Label 1-6048-6048.1957-r0-0-0
Degree $1$
Conductor $6048$
Sign $-0.157 - 0.987i$
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 + (−0.906 + 0.422i)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 + (0.422 − 0.906i)29-s + (−0.939 − 0.342i)31-s + (0.707 + 0.707i)37-s + (−0.342 + 0.939i)41-s + (−0.819 + 0.573i)43-s + (0.939 − 0.342i)47-s + (−0.258 + 0.965i)53-s − i·55-s + ⋯
L(s)  = 1  + (−0.422 − 0.906i)5-s + (0.906 + 0.422i)11-s + (−0.906 + 0.422i)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 + (0.422 − 0.906i)29-s + (−0.939 − 0.342i)31-s + (0.707 + 0.707i)37-s + (−0.342 + 0.939i)41-s + (−0.819 + 0.573i)43-s + (0.939 − 0.342i)47-s + (−0.258 + 0.965i)53-s − i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.157 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.157 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $-0.157 - 0.987i$
Analytic conductor: \(28.0867\)
Root analytic conductor: \(28.0867\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6048} (1957, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ -0.157 - 0.987i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8360181498 - 0.9796786643i\)
\(L(\frac12)\) \(\approx\) \(0.8360181498 - 0.9796786643i\)
\(L(1)\) \(\approx\) \(0.9373803003 - 0.2174133696i\)
\(L(1)\) \(\approx\) \(0.9373803003 - 0.2174133696i\)

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 \( 1 + (-0.906 + 0.422i)T \)
17 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (-0.965 - 0.258i)T \)
23 \( 1 + (0.984 - 0.173i)T \)
29 \( 1 + (0.422 - 0.906i)T \)
31 \( 1 + (-0.939 - 0.342i)T \)
37 \( 1 + (0.707 + 0.707i)T \)
41 \( 1 + (-0.342 + 0.939i)T \)
43 \( 1 + (-0.819 + 0.573i)T \)
47 \( 1 + (0.939 - 0.342i)T \)
53 \( 1 + (-0.258 + 0.965i)T \)
59 \( 1 + (0.996 + 0.0871i)T \)
61 \( 1 + (0.422 - 0.906i)T \)
67 \( 1 + (-0.819 - 0.573i)T \)
71 \( 1 + (0.866 - 0.5i)T \)
73 \( 1 - iT \)
79 \( 1 + (-0.173 + 0.984i)T \)
83 \( 1 + (-0.422 + 0.906i)T \)
89 \( 1 + (0.866 - 0.5i)T \)
97 \( 1 + (0.173 - 0.984i)T \)
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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

−17.77120250695943574157139603870, −17.258719581476177883894960922685, −16.65072213380793319828752938665, −15.90009436227457387870331946756, −15.04926959864811694158437497870, −14.54095523300453988522401698654, −14.37110214535668809778187770037, −13.1874259199963882536505912482, −12.60201510522204273378914953040, −11.911805135739029479341203592258, −11.28167519915164044941795684216, −10.46984603077215350881508463387, −10.24077862006747311862876355531, −9.10550129719618164091804139191, −8.59948292720566505926069863837, −7.72499793510793097079697903131, −7.09335827698621757414268364774, −6.54491999679325945897186562362, −5.74829629911124269192607086697, −5.007243102806185434137030743616, −3.89970524880950834255189492905, −3.60446074631012143647737048819, −2.69218763550566040509532023991, −1.932129313447729134287161769366, −0.88161418489846142868048336395, 0.397375880719767083007954370100, 1.29265729932562987792936663045, 2.13338845980892130416705680564, 3.016040695791336509932232463113, 3.97816920879397792837284511917, 4.63465459083498431594155583030, 5.00674206904719863528987915283, 6.0505695206545019312711164812, 6.82526423114553976749094565684, 7.44539422873232642009049898449, 8.17899994829796715080969471806, 8.942786048880798778066006614274, 9.48634677756836529504539645438, 9.994094655160918961783902382434, 11.16179604228174343758782122404, 11.638473883719156528880961315336, 12.26706258108330559099616180746, 12.81094969449263089892010584367, 13.52590138616755750249017735965, 14.31765255916816834810652587678, 15.027412492346671469482791473898, 15.40683145173155253868358800031, 16.52258272917506549319614406563, 16.80946103703364823460607198338, 17.22959334840261832727191850495

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