Properties

Label 1-6048-6048.5717-r0-0-0
Degree $1$
Conductor $6048$
Sign $0.162 + 0.986i$
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  + (−0.0871 − 0.996i)5-s + (−0.996 − 0.0871i)11-s + (0.996 − 0.0871i)13-s + (0.5 + 0.866i)17-s + (0.258 + 0.965i)19-s + (−0.342 − 0.939i)23-s + (−0.984 + 0.173i)25-s + (−0.0871 + 0.996i)29-s + (−0.766 − 0.642i)31-s + (−0.707 − 0.707i)37-s + (0.642 − 0.766i)41-s + (−0.422 − 0.906i)43-s + (−0.766 + 0.642i)47-s + (0.965 − 0.258i)53-s i·55-s + ⋯
L(s)  = 1  + (−0.0871 − 0.996i)5-s + (−0.996 − 0.0871i)11-s + (0.996 − 0.0871i)13-s + (0.5 + 0.866i)17-s + (0.258 + 0.965i)19-s + (−0.342 − 0.939i)23-s + (−0.984 + 0.173i)25-s + (−0.0871 + 0.996i)29-s + (−0.766 − 0.642i)31-s + (−0.707 − 0.707i)37-s + (0.642 − 0.766i)41-s + (−0.422 − 0.906i)43-s + (−0.766 + 0.642i)47-s + (0.965 − 0.258i)53-s i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.162 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.162 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $0.162 + 0.986i$
Analytic conductor: \(28.0867\)
Root analytic conductor: \(28.0867\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6048} (5717, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ 0.162 + 0.986i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6617178724 + 0.5617178859i\)
\(L(\frac12)\) \(\approx\) \(0.6617178724 + 0.5617178859i\)
\(L(1)\) \(\approx\) \(0.9129012155 - 0.06912518063i\)
\(L(1)\) \(\approx\) \(0.9129012155 - 0.06912518063i\)

Euler product

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

−17.92188813031832564809031058741, −16.88034493275412470478682716569, −16.05684638441553794893275969713, −15.57764617706136060733027042834, −15.09431903701861416756655833162, −14.19281983867658106868104796926, −13.59035269041165011880645096139, −13.22792451510646999385096676526, −12.17698291827473740916404371588, −11.419334836086641505270143839615, −11.06908197106475492694946431774, −10.28148823020032144570802122909, −9.6795643321005365646953637390, −8.96571222830784939491853285368, −7.87952584983478870269044890602, −7.65227425417090310421309727845, −6.73338506192979894698275642460, −6.16706060522249052850979550297, −5.31314413750779219562560390355, −4.68803486457350014609078960623, −3.52479629049486438457717269540, −3.1760176885109729755816759334, −2.35039506520995254445737930618, −1.48578240583872113321840244804, −0.23756008843350668434030165015, 0.95513887985740601048210483702, 1.66960033263249029988077345523, 2.523405944708493509384835008412, 3.72872063672948720366360512374, 3.93860192771959034666889698566, 5.097961455338736761971676247572, 5.59330860875158438325601672460, 6.1400652566044722821404347783, 7.262878581599517331147914900341, 7.93652983110130015634128840722, 8.52931624804975719810813393925, 8.97785679927406108182139871071, 10.01382959507543696015181437483, 10.51117657959507423059366193632, 11.18191653671741731238827739293, 12.152494641728200398234907732352, 12.61928927405668612111099787028, 13.106984971727067834977301893522, 13.86962530208586671430202360051, 14.549024419874956988235470430808, 15.36738776748736533498094183436, 16.05459276989923249417320885001, 16.44135168928771271919842201609, 17.03551030608620807063892693921, 17.95711294091120543436241613799

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