Properties

Degree 1
Conductor $ 13 \cdot 31 $
Sign $0.612 - 0.790i$
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 − 6-s − 7-s − 8-s + (−0.5 − 0.866i)9-s + 10-s − 11-s + (−0.5 − 0.866i)12-s + (−0.5 − 0.866i)14-s + (0.5 + 0.866i)15-s + (−0.5 − 0.866i)16-s + 17-s + (0.5 − 0.866i)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 − 6-s − 7-s − 8-s + (−0.5 − 0.866i)9-s + 10-s − 11-s + (−0.5 − 0.866i)12-s + (−0.5 − 0.866i)14-s + (0.5 + 0.866i)15-s + (−0.5 − 0.866i)16-s + 17-s + (0.5 − 0.866i)18-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.612 - 0.790i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\n\]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 403 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.612 - 0.790i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\n\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(403\)    =    \(13 \cdot 31\)
\( \varepsilon \)  =  $0.612 - 0.790i$
motivic weight  =  \(0\)
character  :  $\chi_{403} (160, \cdot )$
Sato-Tate  :  $\mu(6)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 403,\ (0:\ ),\ 0.612 - 0.790i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.3295697727 - 0.1616258394i$
$L(\frac12,\chi)$  $\approx$  $0.3295697727 - 0.1616258394i$
$L(\chi,1)$  $\approx$  0.6745583243 + 0.3490207563i
$L(1,\chi)$  $\approx$  0.6745583243 + 0.3490207563i

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

−24.08048231949950768159152361044, −23.35570190895095349184341187794, −22.78938813860376708789025159010, −21.95747869099141684316136473155, −21.254133820269713058185112046332, −20.00777153039758507256054592270, −19.0695890134764693357107482217, −18.64433551076233589125959921029, −17.87300405173841744773670069623, −16.77148040044618274334689055839, −15.50172379792183800033748497693, −14.42633241375039216432194238940, −13.509403469828942530040469186277, −12.94533459032177171863303485827, −12.12712819294649769598118714329, −11.02358542558130956137127461329, −10.353174544688779152983414245720, −9.51863743417569724463216202198, −7.95190224642717006011369702755, −6.73278074548821838450745702819, −5.98128760552066536218061070948, −5.1719313299911964676083314282, −3.45138359695777361801816599283, −2.65127800967674303620394796128, −1.61227031462137361457238410216, 0.18356799175620287330425139792, 2.7169468300158557994339110208, 3.93533863778720575429263504183, 4.80309481372466431812823330832, 5.78244026689106348779071504906, 6.26575012759791486160466614113, 7.78470940905476055000555711950, 8.8437598810404689188251300019, 9.66845218676592173589295548995, 10.518389194938295689740596197, 12.11484742712130504739597053963, 12.71505194942792450413142206925, 13.56372107537433695264543081519, 14.71134754312560584936961820047, 15.63591159360057595941961336635, 16.36782836854120830704313078199, 16.79636982307017113023955276836, 17.71477202411155919567895619272, 18.78687839919128590150336728373, 20.300692239457487505020976553362, 21.11077728366650039143993390277, 21.63857999445813119278020105769, 22.667334735588852173996504403735, 23.29377158144323314186484298386, 24.071630890196555726641559158931

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