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Properties

Label 1-1148-1148.67-r0-0-0
Degree $1$
Conductor $1148$
Sign $0.187 - 0.982i$
Analytic cond. $5.33128$
Root an. cond. $5.33128$
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  + (−0.965 − 0.258i)3-s + (−0.406 − 0.913i)5-s + (0.866 + 0.5i)9-s + (−0.933 + 0.358i)11-s + (0.453 + 0.891i)13-s + (0.156 + 0.987i)15-s + (−0.358 − 0.933i)17-s + (−0.544 − 0.838i)19-s + (0.669 + 0.743i)23-s + (−0.669 + 0.743i)25-s + (−0.707 − 0.707i)27-s + (0.987 − 0.156i)29-s + (0.913 + 0.406i)31-s + (0.994 − 0.104i)33-s + (0.913 − 0.406i)37-s + ⋯
L(s)  = 1  + (−0.965 − 0.258i)3-s + (−0.406 − 0.913i)5-s + (0.866 + 0.5i)9-s + (−0.933 + 0.358i)11-s + (0.453 + 0.891i)13-s + (0.156 + 0.987i)15-s + (−0.358 − 0.933i)17-s + (−0.544 − 0.838i)19-s + (0.669 + 0.743i)23-s + (−0.669 + 0.743i)25-s + (−0.707 − 0.707i)27-s + (0.987 − 0.156i)29-s + (0.913 + 0.406i)31-s + (0.994 − 0.104i)33-s + (0.913 − 0.406i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.187 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.187 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $0.187 - 0.982i$
Analytic conductor: \(5.33128\)
Root analytic conductor: \(5.33128\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1148,\ (0:\ ),\ 0.187 - 0.982i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6052160561 - 0.5005290239i\)
\(L(\frac12)\) \(\approx\) \(0.6052160561 - 0.5005290239i\)
\(L(1)\) \(\approx\) \(0.6872640272 - 0.1778707866i\)
\(L(1)\) \(\approx\) \(0.6872640272 - 0.1778707866i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
41 \( 1 \)
good3 \( 1 + (-0.965 - 0.258i)T \)
5 \( 1 + (-0.406 - 0.913i)T \)
11 \( 1 + (-0.933 + 0.358i)T \)
13 \( 1 + (0.453 + 0.891i)T \)
17 \( 1 + (-0.358 - 0.933i)T \)
19 \( 1 + (-0.544 - 0.838i)T \)
23 \( 1 + (0.669 + 0.743i)T \)
29 \( 1 + (0.987 - 0.156i)T \)
31 \( 1 + (0.913 + 0.406i)T \)
37 \( 1 + (0.913 - 0.406i)T \)
43 \( 1 + (-0.951 - 0.309i)T \)
47 \( 1 + (-0.0523 + 0.998i)T \)
53 \( 1 + (0.629 + 0.777i)T \)
59 \( 1 + (0.978 - 0.207i)T \)
61 \( 1 + (0.207 - 0.978i)T \)
67 \( 1 + (-0.777 + 0.629i)T \)
71 \( 1 + (-0.156 + 0.987i)T \)
73 \( 1 + (0.866 - 0.5i)T \)
79 \( 1 + (-0.258 - 0.965i)T \)
83 \( 1 - T \)
89 \( 1 + (0.838 - 0.544i)T \)
97 \( 1 + (-0.156 - 0.987i)T \)
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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

−21.45283886346454250222362219774, −21.00776150346666770578660024151, −19.85818493260752648110545913949, −18.92764754846644007765075099763, −18.31887604874463901765273860295, −17.74648458984019780770739456165, −16.78306970530233839328307579425, −16.063213541937557218612575307685, −15.19264170396047774175900375994, −14.87155947196504143078877122022, −13.47829395120720204698921118834, −12.810922621948333925605174217586, −11.90703227982872872325214396475, −11.09071494600666246994370776072, −10.37254682767199611544999217702, −10.153009536025577212216514410121, −8.47692065109580602084128262249, −7.90119957896295510420578099471, −6.70071694152199161371880623556, −6.190289981361698996820344469952, −5.28870950487967988820363015296, −4.26917263045734900963840449377, −3.40255228403499201522843207216, −2.40321276130356304032647554174, −0.86099007496522346872944773972, 0.51215424608903470335187233998, 1.55737068673088592623143129094, 2.74385462606651636243316641356, 4.319078021101032074211870504551, 4.75118227483363161620111723346, 5.56394540195580518430251266524, 6.65391510864471529309329110415, 7.33622032868072673903663868595, 8.300720523776106089597264702630, 9.191574839970587240619475274861, 10.0829347784439424247685288801, 11.161566842418652293057155969913, 11.60782855337498717223104374387, 12.47774567628332673571525150849, 13.197984362412466345768600728952, 13.76222645877237599720021852523, 15.28605857152217216109810176469, 15.87416215456040857654943372659, 16.41167261678282237267317386656, 17.345158939632424959029630126953, 17.87113623032577203137644786410, 18.81396344684840130684684796150, 19.47122016811564184788765103515, 20.441803022668964812852127780209, 21.247385996574425229242294982834

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