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Properties

Label 1-83-83.53-r1-0-0
Degree $1$
Conductor $83$
Sign $0.323 + 0.946i$
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  + (−0.190 − 0.981i)2-s + (0.338 − 0.941i)3-s + (−0.927 + 0.373i)4-s + (−0.896 + 0.443i)5-s + (−0.988 − 0.152i)6-s + (0.0383 − 0.999i)7-s + (0.543 + 0.839i)8-s + (−0.771 − 0.636i)9-s + (0.606 + 0.795i)10-s + (−0.114 + 0.993i)11-s + (0.0383 + 0.999i)12-s + (−0.817 + 0.575i)13-s + (−0.988 + 0.152i)14-s + (0.114 + 0.993i)15-s + (0.720 − 0.693i)16-s + (−0.264 + 0.964i)17-s + ⋯
L(s)  = 1  + (−0.190 − 0.981i)2-s + (0.338 − 0.941i)3-s + (−0.927 + 0.373i)4-s + (−0.896 + 0.443i)5-s + (−0.988 − 0.152i)6-s + (0.0383 − 0.999i)7-s + (0.543 + 0.839i)8-s + (−0.771 − 0.636i)9-s + (0.606 + 0.795i)10-s + (−0.114 + 0.993i)11-s + (0.0383 + 0.999i)12-s + (−0.817 + 0.575i)13-s + (−0.988 + 0.152i)14-s + (0.114 + 0.993i)15-s + (0.720 − 0.693i)16-s + (−0.264 + 0.964i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.323 + 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.323 + 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(83\)
Sign: $0.323 + 0.946i$
Analytic conductor: \(8.91958\)
Root analytic conductor: \(8.91958\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{83} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 83,\ (1:\ ),\ 0.323 + 0.946i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03634360428 + 0.02597644584i\)
\(L(\frac12)\) \(\approx\) \(0.03634360428 + 0.02597644584i\)
\(L(1)\) \(\approx\) \(0.4931079220 - 0.3825728961i\)
\(L(1)\) \(\approx\) \(0.4931079220 - 0.3825728961i\)

Euler product

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

−31.18590634802787455793310558730, −28.95535599920062084731124225733, −27.69930579039576542203047961980, −27.24375639098639353206802994390, −26.318532481053968386982002418112, −24.9163040843161115814678408989, −24.39492039940722105859716298581, −22.800072073124136278300782452307, −22.09590244289764406175123144990, −20.69154184357421645143273184392, −19.41023713001642311711342246107, −18.415578223602339437654565359095, −16.725626192025400990662278642502, −16.05016606222310892959185039893, −15.16818996852811801968896062876, −14.26886663323109589160616319982, −12.626774425175497353663313993217, −11.09362625856823330957556119197, −9.49582516177692915645970961488, −8.62761250238045651514934541372, −7.67433335861818951122313009343, −5.65960944383166677756489855862, −4.756029425805893103391751789956, −3.23161072523723009026952066463, −0.02171542601507234150847120667, 1.70219832464217839894271249808, 3.22325659006092448169596327381, 4.50737205597567693373492875341, 7.07225723442242884262223016985, 7.75428169021068877663700805889, 9.27433457605766934975185607496, 10.728478357444404921554554758773, 11.80237323189217386574649928801, 12.78719723180784362549030645096, 13.90502112745341403661108667552, 15.02915799209978187496400806781, 17.11754870523789957792518666185, 17.92059520726760648340700502240, 19.36109169358742236587290825196, 19.62618040737526435458887132804, 20.71447444142652812671317894305, 22.3370094107207067427848428316, 23.32333527374609555379324656108, 24.05065585625608857202221449341, 25.96981520069900437150050750345, 26.51618563580615449461385875957, 27.75983259666215784239272123969, 28.95197586000417993266403536595, 29.89151345618979349083415753621, 30.78173347844767507742445598490

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