Properties

Degree $1$
Conductor $344$
Sign $0.0861 - 0.996i$
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)3-s + (−0.5 + 0.866i)5-s + (−0.5 − 0.866i)7-s + (−0.5 − 0.866i)9-s + 11-s + (0.5 + 0.866i)13-s + (0.5 + 0.866i)15-s + (−0.5 − 0.866i)17-s + (0.5 − 0.866i)19-s − 21-s + (0.5 − 0.866i)23-s + (−0.5 − 0.866i)25-s − 27-s + (−0.5 − 0.866i)29-s + (0.5 − 0.866i)31-s + ⋯
L(s,χ)  = 1  + (0.5 − 0.866i)3-s + (−0.5 + 0.866i)5-s + (−0.5 − 0.866i)7-s + (−0.5 − 0.866i)9-s + 11-s + (0.5 + 0.866i)13-s + (0.5 + 0.866i)15-s + (−0.5 − 0.866i)17-s + (0.5 − 0.866i)19-s − 21-s + (0.5 − 0.866i)23-s + (−0.5 − 0.866i)25-s − 27-s + (−0.5 − 0.866i)29-s + (0.5 − 0.866i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 344 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.0861 - 0.996i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 344 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.0861 - 0.996i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(344\)    =    \(2^{3} \cdot 43\)
Sign: $0.0861 - 0.996i$
Motivic weight: \(0\)
Character: $\chi_{344} (179, \cdot )$
Sato-Tate group: $\mu(6)$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 344,\ (0:\ ),\ 0.0861 - 0.996i)\)

Particular Values

\(L(\chi,\frac{1}{2})\) \(\approx\) \(0.9342438307 - 0.8569679711i\)
\(L(\frac12,\chi)\) \(\approx\) \(0.9342438307 - 0.8569679711i\)
\(L(\chi,1)\) \(\approx\) \(1.037824855 - 0.3947995588i\)
\(L(1,\chi)\) \(\approx\) \(1.037824855 - 0.3947995588i\)

Euler product

   \(L(\chi,s) = \displaystyle\prod_p (1- \chi(p) p^{-s})^{-1}\)
   \(L(s,\chi) = \displaystyle\prod_p (1- \chi(p) p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−25.058284807424420489646359677607, −24.56274633954021892127389398236, −23.12454891469749669995049563631, −22.39922278161420126152025943543, −21.480901116006424906193792555868, −20.71007853405770779764955372832, −19.66597127491902846610250263254, −19.3920689094566877336285128348, −17.88752370916114019155537680551, −16.77725028839771446561330626600, −16.00597585069072443454323853298, −15.35244974830268925374694932397, −14.50811702241262402101583145006, −13.24413576209867387297408160711, −12.39216895156503573349177626171, −11.40741228533764619948645120223, −10.27471695320333357840942378872, −9.089291962219834581485723007747, −8.77271769566104545415001665637, −7.665490050806906310352216205093, −6.00028295996432403836751121088, −5.151216504103358571133637938190, −3.902062443163436764394017925, −3.22709242053156097184739667087, −1.570916559517032991313955171728, 0.79782100714093808084000850082, 2.35962482748163942178950635823, 3.417966770713466605220784469074, 4.327418397959569597871091225478, 6.45668023727984417768536344593, 6.7748536621614979061725869672, 7.66628331960388158273171268626, 8.89259356701774117637488121854, 9.79878345050977438602318356603, 11.28145826486949931436224970014, 11.69495172146654569274435257950, 13.10793326835084462132749944411, 13.84214073453023808324303990835, 14.51715343583514379313184871455, 15.58408895291233099855582953966, 16.704528507002833673000722683798, 17.69716089363999298954797445451, 18.669124723535035287978639061, 19.32609519238036516471319927935, 19.99926465938060833009641407288, 20.91956583718621385518980756976, 22.53428068028371293119339707841, 22.75172742193526042403141074587, 23.91565675641763797401456911816, 24.54780791271210934912958688699

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