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Properties

Label 1-83-83.42-r1-0-0
Degree $1$
Conductor $83$
Sign $0.996 - 0.0870i$
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  + (0.997 − 0.0765i)2-s + (0.720 + 0.693i)3-s + (0.988 − 0.152i)4-s + (−0.477 − 0.878i)5-s + (0.771 + 0.636i)6-s + (0.817 − 0.575i)7-s + (0.973 − 0.227i)8-s + (0.0383 + 0.999i)9-s + (−0.543 − 0.839i)10-s + (−0.264 − 0.964i)11-s + (0.817 + 0.575i)12-s + (0.927 + 0.373i)13-s + (0.771 − 0.636i)14-s + (0.264 − 0.964i)15-s + (0.953 − 0.301i)16-s + (−0.409 + 0.912i)17-s + ⋯
L(s)  = 1  + (0.997 − 0.0765i)2-s + (0.720 + 0.693i)3-s + (0.988 − 0.152i)4-s + (−0.477 − 0.878i)5-s + (0.771 + 0.636i)6-s + (0.817 − 0.575i)7-s + (0.973 − 0.227i)8-s + (0.0383 + 0.999i)9-s + (−0.543 − 0.839i)10-s + (−0.264 − 0.964i)11-s + (0.817 + 0.575i)12-s + (0.927 + 0.373i)13-s + (0.771 − 0.636i)14-s + (0.264 − 0.964i)15-s + (0.953 − 0.301i)16-s + (−0.409 + 0.912i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.996 - 0.0870i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.996 - 0.0870i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(83\)
Sign: $0.996 - 0.0870i$
Analytic conductor: \(8.91958\)
Root analytic conductor: \(8.91958\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{83} (42, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 83,\ (1:\ ),\ 0.996 - 0.0870i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.840202456 - 0.1675559692i\)
\(L(\frac12)\) \(\approx\) \(3.840202456 - 0.1675559692i\)
\(L(1)\) \(\approx\) \(2.402596793 + 0.03088052139i\)
\(L(1)\) \(\approx\) \(2.402596793 + 0.03088052139i\)

Euler product

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

−30.79712527215908178644607836951, −30.16369500514851202620250065483, −28.872782544128616011636326041917, −27.37981540209007490352593717403, −25.8486255989362975813838825300, −25.31490053747021438412356273708, −24.04659931274902019529915819557, −23.283071016624047691156336378718, −22.17618887558518574199992888643, −20.82106673049098023916676710375, −20.07444165787092707513977073081, −18.68549529336190325222038895110, −17.82035264933825204981406178457, −15.68034408714217751321317805728, −14.93401528739742776513240778298, −14.12303055799551249629385370049, −12.86106802407405760645092693546, −11.82364748488900955458342145059, −10.68351595133405136921964936524, −8.51659767376576744628993694637, −7.37810565992567042494377555411, −6.404388873438888044030701776253, −4.61164435046516979853707518571, −3.07490662899518749421406074060, −2.00487831976325214828280790087, 1.686077153872408123365798856389, 3.6817176786792493706907652804, 4.35971598322255371828364683647, 5.726124835233169649076415367891, 7.77903007110180909047682281087, 8.685301247865897121186546402346, 10.590636301812591390706306101036, 11.43553643588415785132978022880, 13.09297847549956359376559948403, 13.8802912389769602428982584627, 15.06101084757489404834863846984, 16.03497496854141299589038487093, 16.92583815253885418505608103120, 19.21382607215245012665361363030, 20.105663735101229876829592142859, 21.1133967163119807148149512470, 21.52948747188468882126295275813, 23.32653888634109556476216247891, 24.00299347656526887290569324708, 25.063128234169449363439336000814, 26.28711838992642265472552249205, 27.47824065840633142711065359065, 28.43906074396113631786099436638, 29.88661466610117692279259134974, 30.960832068645450248619825885

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