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Properties

Label 1-85-85.23-r0-0-0
Degree $1$
Conductor $85$
Sign $0.940 - 0.340i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.940 - 0.340i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.940 - 0.340i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(85\)    =    \(5 \cdot 17\)
Sign: $0.940 - 0.340i$
Analytic conductor: \(0.394738\)
Root analytic conductor: \(0.394738\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{85} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 85,\ (0:\ ),\ 0.940 - 0.340i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.501091477 - 0.2631715642i\)
\(L(\frac12)\) \(\approx\) \(1.501091477 - 0.2631715642i\)
\(L(1)\) \(\approx\) \(1.525710224 - 0.2290352218i\)
\(L(1)\) \(\approx\) \(1.525710224 - 0.2290352218i\)

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

−30.87712422139981104873630815072, −30.19473093574709110156744568720, −29.06202152282883534604957332993, −27.44851589227244838085762932813, −26.055986206870119875987573857923, −25.48843373930479483091799136060, −24.13619094897060529534994626872, −23.71470130200581180653016699055, −22.65231285955452618147643069948, −20.9482322509620477373591462480, −20.45279161677368304454894012806, −18.55329688786742361136181137012, −17.80829137081405060072653410662, −16.57734438300851248478385570443, −15.11911511688602089976419997843, −14.21613119259142685684588732465, −13.24069993449634742502087014922, −12.28803753435326842102581603641, −10.89897397970350352416248758088, −8.57939862484846384694597903508, −7.84135874347058019396009530823, −6.68181675964789571663247628410, −5.34061663836385866024686857837, −3.76419832603733302869413867023, −2.06187526983940062049471052676, 2.12546154721162117462421908975, 3.51960320186130940922663834947, 4.79368846975929186817080207377, 5.802436911804152658284215225854, 8.16296604441608108520879483008, 9.37896889762086753656117937127, 10.79998806867522949166893918633, 11.344006276474443448587050326540, 13.08356552318137296016829012894, 14.11239157732560538045480587893, 15.18249402699741692941898275785, 15.98060207927602600067977877577, 17.84570488618352013489091757834, 19.08236163749877058483271099862, 20.31121920311471592279419930276, 21.21982723124036983664956009698, 21.670406370401951927979227240323, 23.08959642678864234218189459513, 24.05260885325361216209344476222, 25.394828611829737738649726141525, 26.66339222723310376197979404009, 27.87762852487181916023008340622, 28.34311423678644085855771458282, 29.84790211333728756397624966002, 30.953967668553881168523691010751

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