Properties

Degree 1
Conductor 83
Sign $-0.0583 + 0.998i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.0383 + 0.999i)2-s + (−0.927 + 0.373i)3-s + (−0.997 − 0.0765i)4-s + (0.859 − 0.511i)5-s + (−0.338 − 0.941i)6-s + (0.953 + 0.301i)7-s + (0.114 − 0.993i)8-s + (0.720 − 0.693i)9-s + (0.477 + 0.878i)10-s + (0.606 + 0.795i)11-s + (0.953 − 0.301i)12-s + (−0.190 − 0.981i)13-s + (−0.338 + 0.941i)14-s + (−0.606 + 0.795i)15-s + (0.988 + 0.152i)16-s + (−0.543 + 0.839i)17-s + ⋯
L(s,χ)  = 1  + (−0.0383 + 0.999i)2-s + (−0.927 + 0.373i)3-s + (−0.997 − 0.0765i)4-s + (0.859 − 0.511i)5-s + (−0.338 − 0.941i)6-s + (0.953 + 0.301i)7-s + (0.114 − 0.993i)8-s + (0.720 − 0.693i)9-s + (0.477 + 0.878i)10-s + (0.606 + 0.795i)11-s + (0.953 − 0.301i)12-s + (−0.190 − 0.981i)13-s + (−0.338 + 0.941i)14-s + (−0.606 + 0.795i)15-s + (0.988 + 0.152i)16-s + (−0.543 + 0.839i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.0583 + 0.998i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.0583 + 0.998i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(83\)
\( \varepsilon \)  =  $-0.0583 + 0.998i$
motivic weight  =  \(0\)
character  :  $\chi_{83} (74, \cdot )$
Sato-Tate  :  $\mu(82)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 83,\ (1:\ ),\ -0.0583 + 0.998i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.9736711696 + 1.032287978i$
$L(\frac12,\chi)$  $\approx$  $0.9736711696 + 1.032287978i$
$L(\chi,1)$  $\approx$  0.8444519999 + 0.5336425596i
$L(1,\chi)$  $\approx$  0.8444519999 + 0.5336425596i

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

−30.24304132519713861373844594650, −29.16815432873781547020084620212, −28.67548907340244407900028211210, −27.23981851077374845977338789402, −26.6162659238258897628349434245, −24.73921564332309165598843559167, −23.78276819316075975704140953424, −22.438001921849801123297857057256, −21.8181763789970734945565240401, −20.83718928973945701433557486299, −19.3259071412542728143587779001, −18.19250914309503320943011087047, −17.65375405272611375410515191449, −16.49997292245558312339135388225, −14.13586995187830349607754476483, −13.721241189312297655623860448456, −12.02426071886270424958175761175, −11.26294765035277528765560974370, −10.316733127548200994146186690096, −8.940005307978344394560580558089, −7.11970450110066394569352262682, −5.63184206361034076786709397728, −4.37767934117231338072907311769, −2.301847070564057624969875987211, −0.997568451259086864001704780600, 1.26125769337433615138948726785, 4.38102446057639176067836816255, 5.33706726363853667732108624477, 6.24887906709150269282566114850, 7.83780756097675535305703157570, 9.2953329708330364818985646679, 10.25779523687714464958617797565, 11.979527281674859209758687358876, 13.14159492612447100732960074665, 14.588865435146564064591058310567, 15.533287374145138495020672664643, 16.82103407196280746858261219902, 17.67154963151698882618233386052, 18.06932797785536338968077001244, 20.25411806933089721105828760859, 21.657543318202960007060170119130, 22.28560440149444267094487116871, 23.57670973557199867242958249645, 24.5244591698149507498434724411, 25.31105482755043329737164435273, 26.71683137173579775927714075926, 27.82868896234705477916502247183, 28.27704907180508682843739867501, 29.76621291879050016345112195155, 31.053377004240308885424934893818

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