↑

Properties

Label 1-83-83.71-r1-0-0
Degree $1$
Conductor $83$
Sign $0.999 - 0.0435i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.817 + 0.575i)2-s + (0.988 − 0.152i)3-s + (0.338 − 0.941i)4-s + (0.665 − 0.746i)5-s + (−0.720 + 0.693i)6-s + (0.190 + 0.981i)7-s + (0.264 + 0.964i)8-s + (0.953 − 0.301i)9-s + (−0.114 + 0.993i)10-s + (−0.543 − 0.839i)11-s + (0.190 − 0.981i)12-s + (0.997 − 0.0765i)13-s + (−0.720 − 0.693i)14-s + (0.543 − 0.839i)15-s + (−0.771 − 0.636i)16-s + (−0.973 − 0.227i)17-s + ⋯
L(s)  = 1  + (−0.817 + 0.575i)2-s + (0.988 − 0.152i)3-s + (0.338 − 0.941i)4-s + (0.665 − 0.746i)5-s + (−0.720 + 0.693i)6-s + (0.190 + 0.981i)7-s + (0.264 + 0.964i)8-s + (0.953 − 0.301i)9-s + (−0.114 + 0.993i)10-s + (−0.543 − 0.839i)11-s + (0.190 − 0.981i)12-s + (0.997 − 0.0765i)13-s + (−0.720 − 0.693i)14-s + (0.543 − 0.839i)15-s + (−0.771 − 0.636i)16-s + (−0.973 − 0.227i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 - 0.0435i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 - 0.0435i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(83\)
Sign: $0.999 - 0.0435i$
Analytic conductor: \(8.91958\)
Root analytic conductor: \(8.91958\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{83} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 83,\ (1:\ ),\ 0.999 - 0.0435i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.845396796 - 0.04017762826i\)
\(L(\frac12)\) \(\approx\) \(1.845396796 - 0.04017762826i\)
\(L(1)\) \(\approx\) \(1.224417949 + 0.06012399817i\)
\(L(1)\) \(\approx\) \(1.224417949 + 0.06012399817i\)

Euler product

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

Imaginary part of the first few zeros on the critical line

−30.64112763735560558915453765712, −29.62793955856106600134247977807, −28.57340372067538414238442106708, −27.150744576564759445586190143446, −26.32294283278012346418078709205, −25.80657010439347773878406267090, −24.68576465299257870437662741215, −22.905014143009776879861068848203, −21.57055555603573998329486122488, −20.60483696095961875560052958842, −19.98744582653741407086152665222, −18.5461578684621818368155004157, −17.932229278957131479256766170388, −16.49705145521935732777824984698, −15.14804241392125457818106727265, −13.79782627684945663351111464645, −12.95080325139634085425759080600, −10.93964127933612537827510778921, −10.22932217279481341982399055433, −9.14685330250366035111635853564, −7.78754276724781099878616743478, −6.82341231583079364717176685956, −4.160647305869158527474217133362, −2.841698753407207586513446350830, −1.56369188791590384541904951401, 1.23829226838709331568735529246, 2.66037645299522553307432660331, 5.104498880097944602861341311623, 6.348089858265434227716709507897, 8.06182070281871615478083586617, 8.82527114376959537199329041886, 9.59484584271157954999451562602, 11.2606032632179997675992270308, 13.13715313959910071584073758847, 13.99614352191356596486177030859, 15.48271338857486248411323003763, 16.068283908919812913954912277140, 17.765144006890767088004129191860, 18.46133087569161475952223613366, 19.626104031770078460551478255181, 20.71088384818782227214799780480, 21.62285375057593378634964515449, 23.71504725805816261134207236574, 24.67086412053468058553198512297, 25.19451571641023681113181492251, 26.20808277117361356205525063403, 27.23463861327572396585953548860, 28.48357817304348111207713900918, 29.16869820441072009444269695449, 30.76679665887194333817599303717

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