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Properties

Label 1-85-85.12-r0-0-0
Degree $1$
Conductor $85$
Sign $0.713 - 0.700i$
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 + (−0.923 − 0.382i)3-s + i·4-s + (0.382 + 0.923i)6-s + (0.382 + 0.923i)7-s + (0.707 − 0.707i)8-s + (0.707 + 0.707i)9-s + (−0.382 − 0.923i)11-s + (0.382 − 0.923i)12-s + 13-s + (0.382 − 0.923i)14-s − 16-s − i·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 + (0.382 + 0.923i)6-s + (0.382 + 0.923i)7-s + (0.707 − 0.707i)8-s + (0.707 + 0.707i)9-s + (−0.382 − 0.923i)11-s + (0.382 − 0.923i)12-s + 13-s + (0.382 − 0.923i)14-s − 16-s − i·18-s + (0.707 − 0.707i)19-s − i·21-s + (−0.382 + 0.923i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.713 - 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.713 - 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(85\)    =    \(5 \cdot 17\)
Sign: $0.713 - 0.700i$
Analytic conductor: \(0.394738\)
Root analytic conductor: \(0.394738\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{85} (12, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 85,\ (0:\ ),\ 0.713 - 0.700i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5215389013 - 0.2133834234i\)
\(L(\frac12)\) \(\approx\) \(0.5215389013 - 0.2133834234i\)
\(L(1)\) \(\approx\) \(0.6011742621 - 0.1924194575i\)
\(L(1)\) \(\approx\) \(0.6011742621 - 0.1924194575i\)

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 \( 1 + (-0.382 - 0.923i)T \)
13 \( 1 + T \)
19 \( 1 + (0.707 - 0.707i)T \)
23 \( 1 + (0.923 - 0.382i)T \)
29 \( 1 + (0.923 + 0.382i)T \)
31 \( 1 + (-0.382 + 0.923i)T \)
37 \( 1 + (0.923 + 0.382i)T \)
41 \( 1 + (0.923 - 0.382i)T \)
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 \( 1 + (-0.923 + 0.382i)T \)
67 \( 1 - iT \)
71 \( 1 + (0.382 - 0.923i)T \)
73 \( 1 + (0.382 - 0.923i)T \)
79 \( 1 + (0.382 + 0.923i)T \)
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

−30.839370116644491379897794489476, −29.42912261401407312698898041136, −28.54550739829878223662882468912, −27.60679441108256117994942862848, −26.79467199218394712046234666473, −25.78508244013561002058414484277, −24.469570560281909735320305644258, −23.20683656386648775372786583906, −23.01406296500408447883448587307, −21.08735843371023400057948478881, −20.11848484743831906115708538520, −18.47127964031954425667078671436, −17.69879044098570366996275649436, −16.756656904399591788554622931385, −15.8366825957780587115572778372, −14.730909192568542955330673131388, −13.26657279298477246594779932541, −11.46605485186058825954411058441, −10.49502087754112214239328897365, −9.56410064077261339014903834441, −7.86084223970790546274183702489, −6.77668088134026122056043118010, −5.47656917625950576422454747374, −4.253493083343928451797302088053, −1.21907797572128082691403822006, 1.21825552128107425925997036222, 2.93566143464486909069346730297, 4.967978748354108480638836966580, 6.4217547292743888225678089894, 7.9754493600870317346840584244, 9.07270958588042401346787862689, 10.75049952683947785598325267186, 11.38592817208553022481238548477, 12.48854644691505911906537652193, 13.57109687090249219309077874168, 15.731768485554155249813423249227, 16.61071925003203060022628146480, 18.034935284140605433251945424744, 18.390864283351096184234986028315, 19.55555995936309302397446496312, 21.171484003154008112564927539840, 21.76513031138420406244044749832, 23.00218360651741946775321711206, 24.33123132386203606414935398602, 25.33822320192360899176590724654, 26.750215418646581612280293487240, 27.73241536646318901416619133255, 28.596261988539212699669714779367, 29.23551214559031698132213135434, 30.500054750836169962607748132645

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