Properties

Degree 1
Conductor $ 13 \cdot 79 $
Sign $-0.777 + 0.628i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.5 − 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.5 + 0.866i)4-s + (−0.5 − 0.866i)5-s + (−0.5 + 0.866i)6-s + (−0.5 + 0.866i)7-s + 8-s + (−0.5 + 0.866i)9-s + (−0.5 + 0.866i)10-s + 11-s + 12-s + 14-s + (−0.5 + 0.866i)15-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)17-s + 18-s + ⋯
L(s,χ)  = 1  + (−0.5 − 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.5 + 0.866i)4-s + (−0.5 − 0.866i)5-s + (−0.5 + 0.866i)6-s + (−0.5 + 0.866i)7-s + 8-s + (−0.5 + 0.866i)9-s + (−0.5 + 0.866i)10-s + 11-s + 12-s + 14-s + (−0.5 + 0.866i)15-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)17-s + 18-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 1027 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.777 + 0.628i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 1027 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.777 + 0.628i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(1027\)    =    \(13 \cdot 79\)
\( \varepsilon \)  =  $-0.777 + 0.628i$
motivic weight  =  \(0\)
character  :  $\chi_{1027} (497, \cdot )$
Sato-Tate  :  $\mu(3)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 1027,\ (0:\ ),\ -0.777 + 0.628i)$
$L(\chi,\frac{1}{2})$  $\approx$  $-0.1224247765 - 0.3460310091i$
$L(\frac12,\chi)$  $\approx$  $-0.1224247765 - 0.3460310091i$
$L(\chi,1)$  $\approx$  0.3982294028 - 0.3611193758i
$L(1,\chi)$  $\approx$  0.3982294028 - 0.3611193758i

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

−22.336448599759245745359691794802, −21.589079768157577489988893780345, −20.25630210916230104577083809406, −19.505507240437085350413208833251, −19.07700178115768280729533055330, −17.688660067662221087905626222963, −17.37710686105963878056216013801, −16.59338648833427279704013220645, −15.77506117416946489484004691215, −15.19547027821940489243888871546, −14.47861590129481712700395940914, −13.75450537385609588510651569010, −12.51094951955189075272164502401, −11.24403317503259020666291056965, −10.7233595161925308264772903898, −10.10273856265062282940800979002, −9.14615912158574534400760350908, −8.4386417387556755338742253373, −7.01223653984193377795935117371, −6.77106244031290235836208282510, −5.9193836725232419451495889282, −4.59247688592934330605088643961, −4.06886708323597637974920468449, −3.04703352629600317836274934051, −1.13759011416925982918846057315, 0.24198689204063049279303105195, 1.3386638270328411564498311043, 2.221534293633340079620881718, 3.323198225136784615772899793134, 4.442946947709626187213821713492, 5.348565203547375697869526244328, 6.473316417455602677998072071679, 7.38127089989200515320563240530, 8.39104935237622433738360353821, 8.92779105838621728442400774073, 9.72777445086274085187861344986, 11.01773426527712078255759728512, 11.72329505737781948211751911127, 12.2093401680351459167572737695, 12.85859635320968611729222498690, 13.52965798615151260665866572504, 14.708597058758830762014455544196, 16.06819244425477443746668199337, 16.56358827581496083843388477051, 17.36545637721611073958088902861, 18.093666655487452808566110385692, 19.05310237445316192109143446832, 19.32045442653645406893886630896, 20.12813780712958822518182830890, 20.97921289327010677381371975549

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