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Properties

Label 1-6048-6048.2389-r0-0-0
Degree $1$
Conductor $6048$
Sign $-0.999 + 0.0311i$
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  + (−0.996 − 0.0871i)5-s + (0.0871 + 0.996i)11-s + (0.906 − 0.422i)13-s − 17-s + (−0.707 + 0.707i)19-s + (−0.342 + 0.939i)23-s + (0.984 + 0.173i)25-s + (−0.422 + 0.906i)29-s + (0.173 + 0.984i)31-s + (0.965 − 0.258i)37-s + (−0.342 + 0.939i)41-s + (0.819 − 0.573i)43-s + (−0.173 + 0.984i)47-s + (−0.965 + 0.258i)53-s − i·55-s + ⋯
L(s)  = 1  + (−0.996 − 0.0871i)5-s + (0.0871 + 0.996i)11-s + (0.906 − 0.422i)13-s − 17-s + (−0.707 + 0.707i)19-s + (−0.342 + 0.939i)23-s + (0.984 + 0.173i)25-s + (−0.422 + 0.906i)29-s + (0.173 + 0.984i)31-s + (0.965 − 0.258i)37-s + (−0.342 + 0.939i)41-s + (0.819 − 0.573i)43-s + (−0.173 + 0.984i)47-s + (−0.965 + 0.258i)53-s − i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0311i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0311i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $-0.999 + 0.0311i$
Analytic conductor: \(28.0867\)
Root analytic conductor: \(28.0867\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6048} (2389, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ -0.999 + 0.0311i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.007423423052 + 0.4763797516i\)
\(L(\frac12)\) \(\approx\) \(0.007423423052 + 0.4763797516i\)
\(L(1)\) \(\approx\) \(0.7681626140 + 0.1498230152i\)
\(L(1)\) \(\approx\) \(0.7681626140 + 0.1498230152i\)

Euler product

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

−17.21997891227883290277130363810, −16.710774336673028195507179843991, −15.88251382452707814044528351318, −15.62725265543348687758808223748, −14.80702551878989873373835858030, −14.162008842498368459381257763957, −13.27832200066065046195217210163, −12.97931632129465547152469664374, −11.919568305120257784727034669196, −11.260817712948728132349432432186, −11.09436683637484155073608457045, −10.21624668573553652825829501760, −9.15959041219508407198575238029, −8.61787160929598229124735443225, −8.14700488830149974918970136373, −7.32742893330244194240905019319, −6.410462834782021082061616469472, −6.164526549791477069155658592607, −4.99035516120803925166546230779, −4.09485532568250254475612148673, −3.9218209045509580324712705523, −2.81673875171772731751208113889, −2.17194291285390943778947989942, −0.916047882149647999704457845646, −0.14756922293474290306562259273, 1.201052497216570970864648417641, 1.89142406414369014193137413896, 2.98726290240748133355677128692, 3.67616932953843479996407761237, 4.35922748650697199129862254326, 4.91525066938656015773632061459, 5.95444797481362361970385225533, 6.59606045450236385335315664974, 7.425422433793041917300827667803, 7.908231294602583683920370095466, 8.68762983053843447715460442812, 9.247589220026917350383785107645, 10.165957280564087402796131430327, 10.91438646772186639182113087129, 11.30726761234262336599056758960, 12.232532019780334661556784695311, 12.6694717982310300408775912040, 13.29184912092986691876573536668, 14.223149314258293548404075022221, 14.8751710247087776465267142111, 15.49072923962778278963862220025, 15.92526948360150176269681099892, 16.614625059332466407997527269108, 17.47446146417593939869373128388, 18.00811442675536598360226762849

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