Properties

Degree 1
Conductor $ 2^{6} $
Sign $0.773 - 0.634i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.382 − 0.923i)3-s + (0.923 + 0.382i)5-s + (0.707 + 0.707i)7-s + (−0.707 + 0.707i)9-s + (0.382 − 0.923i)11-s + (0.923 − 0.382i)13-s i·15-s i·17-s + (0.923 − 0.382i)19-s + (0.382 − 0.923i)21-s + (−0.707 + 0.707i)23-s + (0.707 + 0.707i)25-s + (0.923 + 0.382i)27-s + (−0.382 − 0.923i)29-s + 31-s + ⋯
L(s,χ)  = 1  + (−0.382 − 0.923i)3-s + (0.923 + 0.382i)5-s + (0.707 + 0.707i)7-s + (−0.707 + 0.707i)9-s + (0.382 − 0.923i)11-s + (0.923 − 0.382i)13-s i·15-s i·17-s + (0.923 − 0.382i)19-s + (0.382 − 0.923i)21-s + (−0.707 + 0.707i)23-s + (0.707 + 0.707i)25-s + (0.923 + 0.382i)27-s + (−0.382 − 0.923i)29-s + 31-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.773 - 0.634i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 64 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.773 - 0.634i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(64\)    =    \(2^{6}\)
\( \varepsilon \)  =  $0.773 - 0.634i$
motivic weight  =  \(0\)
character  :  $\chi_{64} (59, \cdot )$
Sato-Tate  :  $\mu(16)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 64,\ (1:\ ),\ 0.773 - 0.634i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.685427568 - 0.6030556268i$
$L(\frac12,\chi)$  $\approx$  $1.685427568 - 0.6030556268i$
$L(\chi,1)$  $\approx$  1.221600121 - 0.2742820743i
$L(1,\chi)$  $\approx$  1.221600121 - 0.2742820743i

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

−32.58379253472138807383490546623, −31.00847310088615649815352267021, −29.86104456856111108204980774416, −28.50368400362136428091410540808, −27.94652689042179099381782483093, −26.55920035715278868897837801810, −25.70221558439378000324841207226, −24.26804127993926459196906145132, −23.070939169389033373343993813165, −21.9135506964135906894798131587, −20.83667364701691856018863980077, −20.2336018528982918517420903921, −18.061431484590094130790024917163, −17.22111079768168644084856689127, −16.28091302191520286987042906574, −14.77730978284672137861001539736, −13.77162263996464275611047970936, −12.13395465281575016471633304902, −10.70691159189086348879578866512, −9.80733575096933135088106921701, −8.48932436559246659743990461752, −6.47457800574635351897224364312, −5.097938952697780159861740463712, −3.92992400243022209039916765606, −1.5071200731640421859294593998, 1.26421235038631443804410674542, 2.772595012828704120019151091748, 5.41856088473329152163873747774, 6.23520252452283702491419180922, 7.80897195147453939782264754309, 9.177462698674183606323153391659, 10.99086787290983553485361726266, 11.869537588284717904711183926973, 13.50553301414321407586135724101, 14.10622390628662743146899280660, 15.88665217554722871348645278494, 17.45712896427119018991353075956, 18.13816302468336328779773630220, 19.05607241265494866806698771270, 20.7275066126709011622896244053, 21.9096535487864621531048276792, 22.88030335343523529783719000924, 24.42738552148706173982308687239, 24.88780613073061922891451872641, 26.14466398468374970589878207405, 27.70811869589732717816457673553, 28.729857374124470155130012114, 29.78484701874350587006825727975, 30.482273978115226370429629889285, 31.72412167262569761608345839542

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