Properties

Degree 1
Conductor $ 2^{2} \cdot 3 \cdot 5 $
Sign $1$
Motivic weight 0
Primitive yes
Self-dual yes
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + 7-s + 11-s − 13-s + 17-s − 19-s − 23-s − 29-s − 31-s − 37-s − 41-s + 43-s − 47-s + 49-s + 53-s + 59-s + 61-s + 67-s + 71-s − 73-s + 77-s − 79-s − 83-s − 89-s − 91-s − 97-s + ⋯
L(s,χ)  = 1  + 7-s + 11-s − 13-s + 17-s − 19-s − 23-s − 29-s − 31-s − 37-s − 41-s + 43-s − 47-s + 49-s + 53-s + 59-s + 61-s + 67-s + 71-s − 73-s + 77-s − 79-s − 83-s − 89-s − 91-s − 97-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & \, \Lambda(\chi,1-s) \end{aligned}\n\]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 60 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & \, \Lambda(1-s,\chi) \end{aligned}\n\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{60} (59, \cdot )$
Sato-Tate  :  $\mu(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(1,\ 60,\ (0:\ ),\ 1)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.9618797159$
$L(\frac12,\chi)$  $\approx$  $0.9618797159$
$L(\chi,1)$  $\approx$  1.065554320
$L(1,\chi)$  $\approx$  1.065554320

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−32.58262270514236334702277741913, −31.56412029546067390492273146917, −30.28370271838972914813392774493, −29.56153792303208874429096790028, −27.86125861653568212621604974582, −27.34677171069150831416030737867, −25.89354093744892243070839471337, −24.673927390561852209576229240424, −23.800425575409428011559961836279, −22.36432718487739382101850758381, −21.328431099039773488357789758822, −20.11799276794188069833039241236, −18.9285284080218204999053966311, −17.554960636106842748012592192039, −16.66706135687040887534297750505, −14.9070292412479863596844040316, −14.21207272889969396426157491711, −12.45813508004805044439217178808, −11.39941638239152605493477803422, −9.94252189246792938879975776348, −8.485164567234391764673881624397, −7.166280912138318471719556203672, −5.46144112265697581553500848609, −3.985806388187321605499007281285, −1.88060641687791880942488827805, 1.88060641687791880942488827805, 3.985806388187321605499007281285, 5.46144112265697581553500848609, 7.166280912138318471719556203672, 8.485164567234391764673881624397, 9.94252189246792938879975776348, 11.39941638239152605493477803422, 12.45813508004805044439217178808, 14.21207272889969396426157491711, 14.9070292412479863596844040316, 16.66706135687040887534297750505, 17.554960636106842748012592192039, 18.9285284080218204999053966311, 20.11799276794188069833039241236, 21.328431099039773488357789758822, 22.36432718487739382101850758381, 23.800425575409428011559961836279, 24.673927390561852209576229240424, 25.89354093744892243070839471337, 27.34677171069150831416030737867, 27.86125861653568212621604974582, 29.56153792303208874429096790028, 30.28370271838972914813392774493, 31.56412029546067390492273146917, 32.58262270514236334702277741913

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