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Properties

Label 1-85-85.62-r0-0-0
Degree $1$
Conductor $85$
Sign $-0.296 + 0.955i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.296 + 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.296 + 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(85\)    =    \(5 \cdot 17\)
Sign: $-0.296 + 0.955i$
Analytic conductor: \(0.394738\)
Root analytic conductor: \(0.394738\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{85} (62, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 85,\ (0:\ ),\ -0.296 + 0.955i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4835059596 + 0.6560763378i\)
\(L(\frac12)\) \(\approx\) \(0.4835059596 + 0.6560763378i\)
\(L(1)\) \(\approx\) \(0.7013298420 + 0.5153752784i\)
\(L(1)\) \(\approx\) \(0.7013298420 + 0.5153752784i\)

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

−30.10930920992300185714454190656, −29.68499102492270388028005100563, −28.43042801628379852547250954308, −27.20892570631402607720163995901, −26.47151892910547702551463427197, −25.05434420036120617704310278252, −24.42140524154161508566958267263, −22.93773321246940343105155564654, −21.55021157288570519892875858065, −20.43890626938582189633499939135, −19.560195757485165615218198956725, −18.66915527010179125297751149865, −17.507846827989395203775661419393, −16.84486504699872211126419204871, −14.71249292873629549827927551464, −13.70336601173702337621071676005, −12.307758943115994055210283106955, −11.57262720307467449073060142976, −10.124050870248236340051526870883, −8.651038553636665302868709556093, −7.85100219543112826691617175627, −6.58318633413908131986664177376, −4.23696427884595079105771809307, −2.54966463234721573519823738463, −1.24294292325636769259470645234, 2.1155700899395283929239667386, 4.38966475504819249005653061370, 5.458990980399812362231878925442, 7.199270374118139874358276351620, 8.510912612751181666942814542472, 9.36563237870010346833819800113, 10.52572455438990606368119363759, 11.75729254754939402760649703826, 14.02982498880981606069057375160, 14.82800556849841401731644579653, 15.613283443985138772569646687366, 17.002995512539860710986811831, 17.65563355641682347267423911003, 19.285557309310262857466432685792, 20.07325886174040779399602274119, 21.41271200446541826353570696223, 22.46938509178077259423553198483, 23.940403248702603961205365719388, 24.95534323233665975595164076419, 25.80945417405578065160080724501, 27.00627204165120269058427586556, 27.5937646718533655475754570701, 28.4022438411855266699843966525, 29.9999693058838348746613784287, 31.439642989659104765597170602747

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