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Properties

Label 1-85-85.27-r0-0-0
Degree $1$
Conductor $85$
Sign $0.307 - 0.951i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.307 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.307 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(85\)    =    \(5 \cdot 17\)
Sign: $0.307 - 0.951i$
Analytic conductor: \(0.394738\)
Root analytic conductor: \(0.394738\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{85} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 85,\ (0:\ ),\ 0.307 - 0.951i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.327798810 - 0.9658003181i\)
\(L(\frac12)\) \(\approx\) \(1.327798810 - 0.9658003181i\)
\(L(1)\) \(\approx\) \(1.473158485 - 0.7501629931i\)
\(L(1)\) \(\approx\) \(1.473158485 - 0.7501629931i\)

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

−31.24802822999984717299614475040, −30.18330794128829640588648942663, −29.25392568171396179613232222983, −27.18942611963799027615201922781, −26.52817681883325226196715600071, −25.73481684413007716159391044346, −24.5161254404603392970113679441, −23.74606346614558312702162685246, −22.30419505187416065605404138289, −21.50412095562867417232849334368, −20.325340748723813793213427395609, −19.37355313668420440174909751415, −17.63805923476918361949367472939, −16.322853672151791080966913082193, −15.651471447783206449061243506736, −14.17231109634871721848329881767, −13.74998641991183462635052684025, −12.45731983551322646238575605451, −10.67256547901431183881246679781, −9.239634510145545230399927316, −7.9171163242466080508372545857, −6.98171776166756563953674282668, −5.202421944808067173500022900063, −3.892162293352105031949060949443, −2.806139491209099408946981551979, 2.00223220775164543782063320661, 2.94158764470949418054343373558, 4.50991886356931987137939068419, 6.07598696531026813814081308405, 7.622391639129300560229384519741, 9.29040317057024108173504705095, 10.09004028335683304506272399266, 12.10692010749123264592576960668, 12.55808368191689928885364278718, 13.90243411807633292287147558509, 14.87843971441156617667570364194, 15.748657964298213798429718393, 18.0064291514727372942732855273, 18.91810837060716155704360129418, 19.88369838634679223229504453496, 20.76551918691487595617075869444, 21.88060958239731966044464781625, 22.90228742486148594750288242675, 24.27606509199318988378252793277, 24.98017756106453420783880962062, 26.184004854707881457407834671097, 27.58476106773217820646817251153, 28.74542168315631984754279524086, 29.61678550889126669253307983460, 30.79767243830545073051625942238

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