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Properties

Label 1-85-85.64-r0-0-0
Degree $1$
Conductor $85$
Sign $0.788 - 0.615i$
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  + 2-s − i·3-s + 4-s − i·6-s + i·7-s + 8-s − 9-s − i·11-s − i·12-s − 13-s + i·14-s + 16-s − 18-s − 19-s + 21-s − i·22-s + ⋯
L(s)  = 1  + 2-s − i·3-s + 4-s − i·6-s + i·7-s + 8-s − 9-s − i·11-s − i·12-s − 13-s + i·14-s + 16-s − 18-s − 19-s + 21-s − i·22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.788 - 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.788 - 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(85\)    =    \(5 \cdot 17\)
Sign: $0.788 - 0.615i$
Analytic conductor: \(0.394738\)
Root analytic conductor: \(0.394738\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{85} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 85,\ (0:\ ),\ 0.788 - 0.615i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.555349774 - 0.5352747625i\)
\(L(\frac12)\) \(\approx\) \(1.555349774 - 0.5352747625i\)
\(L(1)\) \(\approx\) \(1.613208594 - 0.3853251154i\)
\(L(1)\) \(\approx\) \(1.613208594 - 0.3853251154i\)

Euler product

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

−31.03164521927013201299338919729, −29.96517445014162253005967155885, −28.89799021648571516600221676938, −27.75828873928685410441008112550, −26.49626168510375771639615317922, −25.62816378870766698346331649043, −24.28927923967664974901902483185, −23.04553760033938023537024737709, −22.46540657117560294358040220526, −21.22178541808547721340378719171, −20.40610347452933828172519107380, −19.562148124909375040537919947672, −17.25192767284111041434740474287, −16.567553454991770460973815966673, −15.15809230999984246180767678372, −14.56405519045038676142775986441, −13.24084780624311489851285848173, −11.93403281815683340987407154425, −10.64882937786606177689341779931, −9.818609691078356618234982720464, −7.791868140992885899779050359968, −6.41606879566465947302021112544, −4.75695524282086629516012628256, −4.138164855376850860902466050781, −2.50406443657761168403028931289, 1.979180325198815572434489359276, 3.17294173168366239038276532750, 5.22745454396544234803046064370, 6.18356011886284214780203225857, 7.43501857103080965194679800935, 8.79624089436864245811951341957, 10.92089968700767657958258260132, 12.047037156453803591963382361, 12.7717628844311640361279946994, 13.98610226832074376325311133742, 14.91804363842413931116419525065, 16.26608203027063150330904767338, 17.58168310085922483001844005916, 19.04801225929277733331817038459, 19.68581743461047671407135194099, 21.304485405554810130643329230759, 22.08211472578180686893455894681, 23.31946281585803207163670973393, 24.26464357747398550538385244297, 24.945045303866794525203757550711, 25.92593561125057726473193926582, 27.74389496511481586320539378760, 29.14690544777684150431973199483, 29.54373544720996682143390170006, 30.79265594661148443102792316420

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