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Properties

Label 1-135-135.4-r0-0-0
Degree $1$
Conductor $135$
Sign $0.448 + 0.893i$
Analytic cond. $0.626937$
Root an. cond. $0.626937$
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  + (−0.766 − 0.642i)2-s + (0.173 + 0.984i)4-s + (−0.173 + 0.984i)7-s + (0.5 − 0.866i)8-s + (−0.939 + 0.342i)11-s + (−0.766 + 0.642i)13-s + (0.766 − 0.642i)14-s + (−0.939 + 0.342i)16-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)19-s + (0.939 + 0.342i)22-s + (−0.173 − 0.984i)23-s + 26-s − 28-s + (0.766 + 0.642i)29-s + ⋯
L(s)  = 1  + (−0.766 − 0.642i)2-s + (0.173 + 0.984i)4-s + (−0.173 + 0.984i)7-s + (0.5 − 0.866i)8-s + (−0.939 + 0.342i)11-s + (−0.766 + 0.642i)13-s + (0.766 − 0.642i)14-s + (−0.939 + 0.342i)16-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)19-s + (0.939 + 0.342i)22-s + (−0.173 − 0.984i)23-s + 26-s − 28-s + (0.766 + 0.642i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.448 + 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.448 + 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(135\)    =    \(3^{3} \cdot 5\)
Sign: $0.448 + 0.893i$
Analytic conductor: \(0.626937\)
Root analytic conductor: \(0.626937\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{135} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 135,\ (0:\ ),\ 0.448 + 0.893i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4704670310 + 0.2901883856i\)
\(L(\frac12)\) \(\approx\) \(0.4704670310 + 0.2901883856i\)
\(L(1)\) \(\approx\) \(0.6349271997 + 0.05192319392i\)
\(L(1)\) \(\approx\) \(0.6349271997 + 0.05192319392i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 + (-0.766 - 0.642i)T \)
7 \( 1 + (-0.173 + 0.984i)T \)
11 \( 1 + (-0.939 + 0.342i)T \)
13 \( 1 + (-0.766 + 0.642i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (-0.173 - 0.984i)T \)
29 \( 1 + (0.766 + 0.642i)T \)
31 \( 1 + (0.173 + 0.984i)T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 + (0.766 - 0.642i)T \)
43 \( 1 + (0.939 - 0.342i)T \)
47 \( 1 + (-0.173 + 0.984i)T \)
53 \( 1 - T \)
59 \( 1 + (-0.939 - 0.342i)T \)
61 \( 1 + (0.173 - 0.984i)T \)
67 \( 1 + (-0.766 + 0.642i)T \)
71 \( 1 + (-0.5 - 0.866i)T \)
73 \( 1 + (0.5 - 0.866i)T \)
79 \( 1 + (0.766 + 0.642i)T \)
83 \( 1 + (-0.766 - 0.642i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
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

−28.1866963927408249121446397442, −27.1655729489506304291770577348, −26.45212283846763727556428296894, −25.57248499313649112337751530627, −24.47691588125953229647461095990, −23.554140177331223184609826527439, −22.78482313773175856414811864409, −21.17770250230250192127534442376, −20.024479421872851758692500826774, −19.27881013468901227320381447214, −18.03042311637183565518874516218, −17.23338198999185571699012639384, −16.21333406604724708995927488640, −15.31399678690972456921954427827, −14.081912482283683423597738981149, −13.112678538785735670031726398, −11.3357050186388350393866434991, −10.310726810707001968447840018471, −9.458428437193344808652568348867, −7.907685933413292286987698585117, −7.30593423880643067560448504860, −5.87730240720007333380610042465, −4.657075779555248873605648684589, −2.670752494917816098146479863787, −0.624401363860935157911849401410, 1.89736296105989094921634827215, 2.979515449248198973589853520610, 4.64218125547109177235279343193, 6.30616732153357820538905387750, 7.77117397151068553993601768719, 8.72223062101175361848704925927, 9.87669584961707024449252207245, 10.77697521444860017401627330212, 12.31968761563055671166525249895, 12.56282351479627844552617834368, 14.37619696340237556926951570237, 15.671601772407983411902365117040, 16.65738977103606529568378482342, 17.79575770070786907315754084327, 18.75349552366143258988903068101, 19.40807728987079316263091754151, 20.758783170743413627478510864342, 21.48560152463135907044651608750, 22.41968723187213632780940527479, 23.82577256721467301683820989289, 25.114728881458448528939488030375, 25.85718564400776450435093753281, 26.851769680719896519386731609631, 27.88072639660221823216453753376, 28.73051263833771822153384870895

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