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Properties

Label 1-85-85.14-r1-0-0
Degree $1$
Conductor $85$
Sign $0.968 + 0.250i$
Analytic cond. $9.13451$
Root an. cond. $9.13451$
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 + (0.923 + 0.382i)3-s − i·4-s + (−0.923 + 0.382i)6-s + (−0.382 − 0.923i)7-s + (0.707 + 0.707i)8-s + (0.707 + 0.707i)9-s + (0.923 − 0.382i)11-s + (0.382 − 0.923i)12-s − i·13-s + (0.923 + 0.382i)14-s − 16-s − 18-s + (0.707 − 0.707i)19-s − i·21-s + (−0.382 + 0.923i)22-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s + (0.923 + 0.382i)3-s − i·4-s + (−0.923 + 0.382i)6-s + (−0.382 − 0.923i)7-s + (0.707 + 0.707i)8-s + (0.707 + 0.707i)9-s + (0.923 − 0.382i)11-s + (0.382 − 0.923i)12-s − i·13-s + (0.923 + 0.382i)14-s − 16-s − 18-s + (0.707 − 0.707i)19-s − i·21-s + (−0.382 + 0.923i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.968 + 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.968 + 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(85\)    =    \(5 \cdot 17\)
Sign: $0.968 + 0.250i$
Analytic conductor: \(9.13451\)
Root analytic conductor: \(9.13451\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{85} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 85,\ (1:\ ),\ 0.968 + 0.250i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.620496926 + 0.2060868501i\)
\(L(\frac12)\) \(\approx\) \(1.620496926 + 0.2060868501i\)
\(L(1)\) \(\approx\) \(1.091463073 + 0.2189730300i\)
\(L(1)\) \(\approx\) \(1.091463073 + 0.2189730300i\)

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 \( 1 + (0.923 + 0.382i)T \)
7 \( 1 + (-0.382 - 0.923i)T \)
11 \( 1 + (0.923 - 0.382i)T \)
13 \( 1 - iT \)
19 \( 1 + (0.707 - 0.707i)T \)
23 \( 1 + (0.923 - 0.382i)T \)
29 \( 1 + (-0.382 + 0.923i)T \)
31 \( 1 + (0.923 + 0.382i)T \)
37 \( 1 + (0.923 + 0.382i)T \)
41 \( 1 + (0.382 + 0.923i)T \)
43 \( 1 + (-0.707 - 0.707i)T \)
47 \( 1 - iT \)
53 \( 1 + (-0.707 + 0.707i)T \)
59 \( 1 + (-0.707 - 0.707i)T \)
61 \( 1 + (-0.382 - 0.923i)T \)
67 \( 1 + T \)
71 \( 1 + (-0.923 - 0.382i)T \)
73 \( 1 + (-0.382 + 0.923i)T \)
79 \( 1 + (0.923 - 0.382i)T \)
83 \( 1 + (0.707 - 0.707i)T \)
89 \( 1 - iT \)
97 \( 1 + (0.382 - 0.923i)T \)
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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

−30.50137809946431807577679705447, −29.347230512710687804304692606335, −28.443636219015225317633302403750, −27.24067786911380157114926879544, −26.26608779827375332313823359348, −25.28131392605584958940622905024, −24.58927237800833062326956830963, −22.69763091512576131073520326916, −21.50023381467940726086165558412, −20.616163287440834185305705265366, −19.35665624053114556991738633945, −18.92017115024898258357114622317, −17.73535013545848440726856623564, −16.34838266138238432248137448924, −14.99354924254588020921706455392, −13.65021053507424701551909866705, −12.42243845807512292983898712950, −11.59176581612154845945933617608, −9.612709026336354630943416879009, −9.15693233760104651333120648813, −7.85224886221858005304273809957, −6.562445535207692616448961634566, −4.04727258395033041439880221760, −2.71743368480998168859905912093, −1.47483216753465449327665106128, 1.03870173904658438839804357666, 3.204096725744895456911218453034, 4.82308053022495394980421490358, 6.64828752562168549730907897057, 7.73614137240779641774672885018, 8.91401043749632349233098804726, 9.905999461304378219325073573910, 10.95711067511358172125007275140, 13.23348094126034354772284250729, 14.204020699161438058609645765165, 15.22318465233910523970906578917, 16.30338754271236377669722668201, 17.26084327709664899062428242478, 18.68049001396662027587529106491, 19.83782811183154708933930346881, 20.27621877178763385621998108624, 22.04633780290280853037513809752, 23.25310497638154336094244995141, 24.612506175870748912618936410381, 25.27905223817930811874722512559, 26.46103776122229073514320960767, 26.98949575585431324936661935824, 28.02097913405855246697954647338, 29.484219225704698038196085904888, 30.50391780400508807247516042834

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