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Properties

Label 1-85-85.8-r1-0-0
Degree $1$
Conductor $85$
Sign $-0.953 - 0.302i$
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  − 2-s + (0.707 − 0.707i)3-s + 4-s + (−0.707 + 0.707i)6-s + (−0.707 − 0.707i)7-s − 8-s − i·9-s + (−0.707 + 0.707i)11-s + (0.707 − 0.707i)12-s − i·13-s + (0.707 + 0.707i)14-s + 16-s + i·18-s + i·19-s − 21-s + (0.707 − 0.707i)22-s + ⋯
L(s)  = 1  − 2-s + (0.707 − 0.707i)3-s + 4-s + (−0.707 + 0.707i)6-s + (−0.707 − 0.707i)7-s − 8-s − i·9-s + (−0.707 + 0.707i)11-s + (0.707 − 0.707i)12-s − i·13-s + (0.707 + 0.707i)14-s + 16-s + i·18-s + i·19-s − 21-s + (0.707 − 0.707i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.953 - 0.302i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.953 - 0.302i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(85\)    =    \(5 \cdot 17\)
Sign: $-0.953 - 0.302i$
Analytic conductor: \(9.13451\)
Root analytic conductor: \(9.13451\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{85} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 85,\ (1:\ ),\ -0.953 - 0.302i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1019751548 - 0.6574042109i\)
\(L(\frac12)\) \(\approx\) \(0.1019751548 - 0.6574042109i\)
\(L(1)\) \(\approx\) \(0.6052421206 - 0.3132633830i\)
\(L(1)\) \(\approx\) \(0.6052421206 - 0.3132633830i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
17 \( 1 \)
good2 \( 1 - T \)
3 \( 1 + (0.707 - 0.707i)T \)
7 \( 1 + (-0.707 - 0.707i)T \)
11 \( 1 + (-0.707 + 0.707i)T \)
13 \( 1 - iT \)
19 \( 1 + iT \)
23 \( 1 + (-0.707 - 0.707i)T \)
29 \( 1 + (-0.707 - 0.707i)T \)
31 \( 1 + (-0.707 - 0.707i)T \)
37 \( 1 + (-0.707 + 0.707i)T \)
41 \( 1 + (0.707 - 0.707i)T \)
43 \( 1 - T \)
47 \( 1 + iT \)
53 \( 1 + T \)
59 \( 1 - iT \)
61 \( 1 + (0.707 - 0.707i)T \)
67 \( 1 - iT \)
71 \( 1 + (-0.707 - 0.707i)T \)
73 \( 1 + (-0.707 + 0.707i)T \)
79 \( 1 + (0.707 - 0.707i)T \)
83 \( 1 + T \)
89 \( 1 + T \)
97 \( 1 + (0.707 - 0.707i)T \)
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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

−31.11348446449084819914552005345, −29.65780615120334436953280973148, −28.56019782266207601782669296010, −27.80117078104750195894559227494, −26.45661875074847956382813163334, −26.07073881705854218219198075218, −25.002178097132380035044038986657, −23.85267978728268045631289648937, −21.86566515079485624735242976534, −21.290206821558504531396569310860, −19.915231776351780449257821545290, −19.17735337234623377494627470805, −18.15534082503904899563747136143, −16.44578411012603351769170705305, −15.93087408980593411351886490242, −14.81456401962598188314890785204, −13.29837989250502891683048601850, −11.62996849504145617217192984057, −10.434349907271904615287959768096, −9.27210458876705458428057530271, −8.62022006976966127685570493523, −7.14928551208293228768263448868, −5.53482336939833596372015828290, −3.42003305007835683183035695176, −2.2114459825474246686601189722, 0.36431188121307968644649447917, 2.07504647399986489416960150773, 3.46330276386227801699323801762, 6.10915973562667917271269341088, 7.388293419073317984310605733632, 8.106342312431641934826166701904, 9.6070875542803056741506112523, 10.465730731332163390030447366499, 12.24640853343987373637936711505, 13.169162238049964782188736486279, 14.71534042615633323957793425872, 15.830137895367347221581111698057, 17.15369846460055847147736068632, 18.218051474065827203494763408378, 19.082404172555052088107468908986, 20.268178361212596739814728413637, 20.63710254812174775214620489451, 22.74445723814943195446488759973, 23.92441210883047329498625422644, 25.03603799908534832156126671801, 25.88144752324669239771897771561, 26.56033053124577829097714977997, 27.86587435414824719647940496623, 29.128827290440336445290234085054, 29.78381336176234019481584283194

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