Properties

Degree 1
Conductor 389
Sign $0.259 - 0.965i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.898 − 0.438i)2-s + (−0.937 + 0.348i)3-s + (0.616 − 0.787i)4-s + (0.925 − 0.378i)5-s + (−0.689 + 0.724i)6-s + (−0.481 + 0.876i)7-s + (0.208 − 0.977i)8-s + (0.756 − 0.653i)9-s + (0.665 − 0.746i)10-s + (−0.852 − 0.523i)11-s + (−0.302 + 0.953i)12-s + (−0.363 − 0.931i)13-s + (−0.0485 + 0.998i)14-s + (−0.735 + 0.677i)15-s + (−0.240 − 0.970i)16-s + (0.665 − 0.746i)17-s + ⋯
L(s,χ)  = 1  + (0.898 − 0.438i)2-s + (−0.937 + 0.348i)3-s + (0.616 − 0.787i)4-s + (0.925 − 0.378i)5-s + (−0.689 + 0.724i)6-s + (−0.481 + 0.876i)7-s + (0.208 − 0.977i)8-s + (0.756 − 0.653i)9-s + (0.665 − 0.746i)10-s + (−0.852 − 0.523i)11-s + (−0.302 + 0.953i)12-s + (−0.363 − 0.931i)13-s + (−0.0485 + 0.998i)14-s + (−0.735 + 0.677i)15-s + (−0.240 − 0.970i)16-s + (0.665 − 0.746i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 389 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.259 - 0.965i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 389 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.259 - 0.965i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(389\)
\( \varepsilon \)  =  $0.259 - 0.965i$
motivic weight  =  \(0\)
character  :  $\chi_{389} (91, \cdot )$
Sato-Tate  :  $\mu(97)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 389,\ (0:\ ),\ 0.259 - 0.965i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(1.406434705 - 1.078951528i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(1.406434705 - 1.078951528i\)
\(L(\chi,1)\)  \(\approx\)  \(1.357194477 - 0.5087862087i\)
\(L(1,\chi)\)  \(\approx\)  \(1.357194477 - 0.5087862087i\)

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

−24.32169022890051402346331348097, −23.71521466229405384702140968345, −22.93403542456918497091171087964, −22.256278519616886892051397576243, −21.44638262646743745081108345979, −20.69118897252885781836533213858, −19.3917864145560555963501441062, −18.23341316556598338337913327633, −17.394651728500901437207534784450, −16.70467166122093177271687950719, −16.02219438973973125423322376721, −14.70416355822332217981956531377, −13.89167019595223466133624603300, −12.962586404395277014276629031149, −12.54247885263504005758556130354, −11.11501600802711937847537602596, −10.54675469170884600335351678320, −9.3695243185654192330916407931, −7.358482452935327537156194017095, −7.16157480368514004988782801409, −6.03195229211545891404958798933, −5.26759361011238563394930093473, −4.26141792460668028413494094881, −2.85036732478453841671589426201, −1.60964532530652900938849459646, 0.925274172553523713852519231678, 2.45941526677532000406350171740, 3.42348051955618861838410129731, 5.07793970093923981020702886751, 5.48673508600827566960266343091, 6.07430526091075911156997135601, 7.471570308791498884983750747945, 9.34848687368424213109391620815, 9.91856026931765465512281999324, 10.86694461592250878194883787954, 11.848062576979613641759335267251, 12.72828572700784309220811196458, 13.22567098001501477559601535731, 14.473826861560029354759355703091, 15.55320271611636942229139831761, 16.134390120965024309868521199665, 17.13051390617821647368668525927, 18.3388126717458221196769018567, 18.856700750960316468414083085987, 20.48297379574197722564931864665, 20.95055690902149635055159443948, 21.85002574268101646419502477255, 22.395284261197498196170169617583, 23.13533684176239104386321191451, 24.23931768774943278098719063357

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