Properties

Degree 1
Conductor 373
Sign $0.887 - 0.461i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.997 + 0.0675i)2-s + (−0.931 − 0.363i)3-s + (0.990 + 0.134i)4-s + (0.409 − 0.912i)5-s + (−0.905 − 0.425i)6-s + (0.688 + 0.724i)7-s + (0.979 + 0.201i)8-s + (0.736 + 0.676i)9-s + (0.470 − 0.882i)10-s + (−0.664 − 0.747i)11-s + (−0.874 − 0.485i)12-s + (0.979 − 0.201i)13-s + (0.638 + 0.769i)14-s + (−0.713 + 0.701i)15-s + (0.963 + 0.266i)16-s + (−0.612 + 0.790i)17-s + ⋯
L(s,χ)  = 1  + (0.997 + 0.0675i)2-s + (−0.931 − 0.363i)3-s + (0.990 + 0.134i)4-s + (0.409 − 0.912i)5-s + (−0.905 − 0.425i)6-s + (0.688 + 0.724i)7-s + (0.979 + 0.201i)8-s + (0.736 + 0.676i)9-s + (0.470 − 0.882i)10-s + (−0.664 − 0.747i)11-s + (−0.874 − 0.485i)12-s + (0.979 − 0.201i)13-s + (0.638 + 0.769i)14-s + (−0.713 + 0.701i)15-s + (0.963 + 0.266i)16-s + (−0.612 + 0.790i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.887 - 0.461i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.887 - 0.461i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(373\)
\( \varepsilon \)  =  $0.887 - 0.461i$
motivic weight  =  \(0\)
character  :  $\chi_{373} (16, \cdot )$
Sato-Tate  :  $\mu(93)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 373,\ (0:\ ),\ 0.887 - 0.461i)$
$L(\chi,\frac{1}{2})$  $\approx$  $2.096527557 - 0.5125493428i$
$L(\frac12,\chi)$  $\approx$  $2.096527557 - 0.5125493428i$
$L(\chi,1)$  $\approx$  1.676432884 - 0.2448737130i
$L(1,\chi)$  $\approx$  1.676432884 - 0.2448737130i

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

−24.26837256760950682542245312632, −23.48794415858673136864740962195, −22.97459380580166938256943044967, −22.19580858187918665938011147075, −21.2360368151248323753118957846, −20.82241374773051313142893572777, −19.64019554017665551501797089790, −18.15851438773612999609476972455, −17.71401862574207747711991036653, −16.583907909803176110519528182746, −15.52704116512181231309258892259, −15.02192897740223575524208873864, −13.7416916858121816561645213833, −13.260716472971474204270716684356, −11.86492970036858966548240933544, −11.02614931024630119340072274466, −10.68069893187597733350992048802, −9.5632075471342368845968023085, −7.53093228727081873431537681437, −6.85483407469975447220703524046, −5.921308498740156238035003729177, −4.86989415651928717336949224380, −4.13512611280622432432553783798, −2.81687317325503503260391064848, −1.48003330251628043228105591736, 1.31083200363004459007029958273, 2.28351511418273497814761040277, 4.02042613304434077530448718388, 5.06406142312021305159161219063, 5.743589661528053263536680238064, 6.36352367372494144579465382127, 7.93049488908223380905533610453, 8.657754173282383001006767467800, 10.562569870745031389914977484922, 11.050980396066340415244835092189, 12.27114259510392373630583883944, 12.713205785999949541224972640803, 13.566758279146836172916864877997, 14.60686465849084373469093539535, 15.99079385067919888239070055790, 16.217580639672536633489466368192, 17.41181065804684022736573380445, 18.20574685575846032122368089952, 19.29491182794072468193122025009, 20.62251232389769321356278907948, 21.30121301044507677172007264880, 21.79998573758171859918395936157, 22.96793622796252205029209814210, 23.683202756457228209752314648362, 24.45523312980745485505815836217

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