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Properties

Label 1-85-85.82-r0-0-0
Degree $1$
Conductor $85$
Sign $-0.721 + 0.691i$
Analytic cond. $0.394738$
Root an. cond. $0.394738$
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  + (0.707 + 0.707i)2-s + (−0.382 + 0.923i)3-s + i·4-s + (−0.923 + 0.382i)6-s + (−0.923 + 0.382i)7-s + (−0.707 + 0.707i)8-s + (−0.707 − 0.707i)9-s + (0.923 − 0.382i)11-s + (−0.923 − 0.382i)12-s + 13-s + (−0.923 − 0.382i)14-s − 16-s − i·18-s + (−0.707 + 0.707i)19-s − i·21-s + (0.923 + 0.382i)22-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)2-s + (−0.382 + 0.923i)3-s + i·4-s + (−0.923 + 0.382i)6-s + (−0.923 + 0.382i)7-s + (−0.707 + 0.707i)8-s + (−0.707 − 0.707i)9-s + (0.923 − 0.382i)11-s + (−0.923 − 0.382i)12-s + 13-s + (−0.923 − 0.382i)14-s − 16-s − i·18-s + (−0.707 + 0.707i)19-s − i·21-s + (0.923 + 0.382i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.721 + 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.721 + 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(85\)    =    \(5 \cdot 17\)
Sign: $-0.721 + 0.691i$
Analytic conductor: \(0.394738\)
Root analytic conductor: \(0.394738\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{85} (82, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 85,\ (0:\ ),\ -0.721 + 0.691i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4180082932 + 1.040165425i\)
\(L(\frac12)\) \(\approx\) \(0.4180082932 + 1.040165425i\)
\(L(1)\) \(\approx\) \(0.8336975523 + 0.8459401743i\)
\(L(1)\) \(\approx\) \(0.8336975523 + 0.8459401743i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
17 \( 1 \)
good2 \( 1 + (0.707 + 0.707i)T \)
3 \( 1 + (-0.382 + 0.923i)T \)
7 \( 1 + (-0.923 + 0.382i)T \)
11 \( 1 + (0.923 - 0.382i)T \)
13 \( 1 + T \)
19 \( 1 + (-0.707 + 0.707i)T \)
23 \( 1 + (0.382 + 0.923i)T \)
29 \( 1 + (0.382 - 0.923i)T \)
31 \( 1 + (0.923 + 0.382i)T \)
37 \( 1 + (0.382 - 0.923i)T \)
41 \( 1 + (0.382 + 0.923i)T \)
43 \( 1 + (0.707 - 0.707i)T \)
47 \( 1 - T \)
53 \( 1 + (-0.707 - 0.707i)T \)
59 \( 1 + (0.707 + 0.707i)T \)
61 \( 1 + (-0.382 - 0.923i)T \)
67 \( 1 - iT \)
71 \( 1 + (-0.923 - 0.382i)T \)
73 \( 1 + (-0.923 - 0.382i)T \)
79 \( 1 + (-0.923 + 0.382i)T \)
83 \( 1 + (0.707 + 0.707i)T \)
89 \( 1 + iT \)
97 \( 1 + (-0.923 - 0.382i)T \)
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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

−30.28481025710108529635631118633, −29.495443847688841230519110377815, −28.550292136375133360755625715006, −27.71375818938129875926378004896, −25.86160175640209501190027289288, −24.804236369083603270466835687485, −23.64097525181190989504828640812, −22.870905437622928571596296900147, −22.094243672631528569195612216614, −20.52453823783990630347309586261, −19.531219347710956526653831193854, −18.78523152547981427114412168130, −17.446305480237019901607550460228, −16.08133454975528569812122150809, −14.51121420627659017073439390416, −13.371600062990085168885587707700, −12.66345146159204603747565036161, −11.52495134310966232103955365133, −10.42874079978438584394379119885, −8.89221131381992961714591060549, −6.81774730954602284152942565569, −6.145127485204572486639190261123, −4.39255405965204270583725931813, −2.84010536791698065157579689095, −1.16090640106352435432527657070, 3.236881047674541608184815230377, 4.18546262301354032322334275675, 5.79880857731885033754044169690, 6.467113960482484473745189301669, 8.46667987789755624581608925581, 9.54292591875581094615126652938, 11.20036471394720563743364890509, 12.2925099506938965854021284849, 13.63237442283956862340670646971, 14.87107171928956619686991183908, 15.86263837815704193139859491334, 16.57419167383338937950084740816, 17.667724027875162990887887862898, 19.365642897760628346414288012061, 20.92552382945578672808707225343, 21.691545849040550168941498087951, 22.73893869931058575061254692749, 23.31189736300713444434349840526, 24.93237663939907670460277198598, 25.73815672586503834389092262075, 26.780161230430245720398822690927, 27.84427415449351002162703124713, 29.08325950409002362779533234595, 30.231206843064180859297512893770, 31.63398534374120821490601845390

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