Properties

Degree 1
Conductor 379
Sign $1$
Motivic weight 0
Primitive yes
Self-dual yes
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  − 2-s − 3-s + 4-s + 5-s + 6-s − 7-s − 8-s + 9-s − 10-s − 11-s − 12-s − 13-s + 14-s − 15-s + 16-s − 17-s − 18-s + 19-s + 20-s + 21-s + 22-s + 23-s + 24-s + 25-s + 26-s − 27-s − 28-s + ⋯
L(s,χ)  = 1  − 2-s − 3-s + 4-s + 5-s + 6-s − 7-s − 8-s + 9-s − 10-s − 11-s − 12-s − 13-s + 14-s − 15-s + 16-s − 17-s − 18-s + 19-s + 20-s + 21-s + 22-s + 23-s + 24-s + 25-s + 26-s − 27-s − 28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 379 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & \, \Lambda(\chi,1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 379 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & \, \Lambda(1-s,\chi) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(379\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{379} (378, \cdot )$
Sato-Tate  :  $\mu(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 379,\ (1:\ ),\ 1)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.5682661129\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.5682661129\)
\(L(\chi,1)\)  \(\approx\)  \(0.4841183254\)
\(L(1,\chi)\)  \(\approx\)  \(0.4841183254\)

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

−24.55566157767619259372651007915, −23.61092924749741532818328746887, −22.3820308354628642211570022891, −21.816973517670714165495845616781, −20.82048004907735304596676239523, −19.85105634669550411838001887778, −18.719580017804240096607874347732, −18.13195322936877710140654937286, −17.331023218310068771611457349939, −16.577439136434493840316855379351, −15.90283561621656083121415190093, −14.869693832128888825189727313498, −13.15171858562111825928069038689, −12.71877323044733548781095949789, −11.39985909082935667398890028901, −10.59121063111293054064315182790, −9.72327289118219699625228889941, −9.25529546686854271211317295340, −7.53807341853369624573845275705, −6.79258886478060940585175967493, −5.872971104654691891251309608280, −5.02290360857028008474067564375, −3.02324158095572599282478064072, −1.94740423360350370719059513847, −0.50538459045467819629987870038, 0.50538459045467819629987870038, 1.94740423360350370719059513847, 3.02324158095572599282478064072, 5.02290360857028008474067564375, 5.872971104654691891251309608280, 6.79258886478060940585175967493, 7.53807341853369624573845275705, 9.25529546686854271211317295340, 9.72327289118219699625228889941, 10.59121063111293054064315182790, 11.39985909082935667398890028901, 12.71877323044733548781095949789, 13.15171858562111825928069038689, 14.869693832128888825189727313498, 15.90283561621656083121415190093, 16.577439136434493840316855379351, 17.331023218310068771611457349939, 18.13195322936877710140654937286, 18.719580017804240096607874347732, 19.85105634669550411838001887778, 20.82048004907735304596676239523, 21.816973517670714165495845616781, 22.3820308354628642211570022891, 23.61092924749741532818328746887, 24.55566157767619259372651007915

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