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Properties

Label 1-83-83.46-r1-0-0
Degree $1$
Conductor $83$
Sign $-0.0583 - 0.998i$
Analytic cond. $8.91958$
Root an. cond. $8.91958$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(83\)
Sign: $-0.0583 - 0.998i$
Analytic conductor: \(8.91958\)
Root analytic conductor: \(8.91958\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{83} (46, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 83,\ (1:\ ),\ -0.0583 - 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9736711696 - 1.032287978i\)
\(L(\frac12)\) \(\approx\) \(0.9736711696 - 1.032287978i\)
\(L(1)\) \(\approx\) \(0.8444519999 - 0.5336425596i\)
\(L(1)\) \(\approx\) \(0.8444519999 - 0.5336425596i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad83 \( 1 \)
good2 \( 1 + (-0.0383 - 0.999i)T \)
3 \( 1 + (-0.927 - 0.373i)T \)
5 \( 1 + (0.859 + 0.511i)T \)
7 \( 1 + (0.953 - 0.301i)T \)
11 \( 1 + (0.606 - 0.795i)T \)
13 \( 1 + (-0.190 + 0.981i)T \)
17 \( 1 + (-0.543 - 0.839i)T \)
19 \( 1 + (0.973 - 0.227i)T \)
23 \( 1 + (-0.665 - 0.746i)T \)
29 \( 1 + (0.896 - 0.443i)T \)
31 \( 1 + (-0.114 + 0.993i)T \)
37 \( 1 + (0.720 - 0.693i)T \)
41 \( 1 + (0.0383 - 0.999i)T \)
43 \( 1 + (0.409 - 0.912i)T \)
47 \( 1 + (0.771 + 0.636i)T \)
53 \( 1 + (0.771 - 0.636i)T \)
59 \( 1 + (-0.264 + 0.964i)T \)
61 \( 1 + (0.817 + 0.575i)T \)
67 \( 1 + (-0.988 + 0.152i)T \)
71 \( 1 + (-0.953 - 0.301i)T \)
73 \( 1 + (-0.477 + 0.878i)T \)
79 \( 1 + (0.997 - 0.0765i)T \)
89 \( 1 + (-0.338 + 0.941i)T \)
97 \( 1 + (-0.338 - 0.941i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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