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Properties

Label 1-85-85.64-r0-0-0
Degree 11
Conductor 8585
Sign 0.788−0.615i0.788 - 0.615i
Analytic cond. 0.3947380.394738
Root an. cond. 0.3947380.394738
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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

Λ(s)=(85s/2 ΓR(s) L(s)=((0.788−0.615i) Λ‾(1−s)\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}
Λ(s)=(85s/2 ΓR(s) L(s)=((0.788−0.615i) Λ‾(1−s)\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: 11
Conductor: 8585    =    5⋅175 \cdot 17
Sign: 0.788−0.615i0.788 - 0.615i
Analytic conductor: 0.3947380.394738
Root analytic conductor: 0.3947380.394738
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ85(64,⋅)\chi_{85} (64, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 85, (0: ), 0.788−0.615i)(1,\ 85,\ (0:\ ),\ 0.788 - 0.615i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.555349774−0.5352747625i1.555349774 - 0.5352747625i
L(12)L(\frac12) ≈\approx 1.555349774−0.5352747625i1.555349774 - 0.5352747625i
L(1)L(1) ≈\approx 1.613208594−0.3853251154i1.613208594 - 0.3853251154i
L(1)L(1) ≈\approx 1.613208594−0.3853251154i1.613208594 - 0.3853251154i

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad5 1 1
17 1 1
good2 1 1
3 1+T 1 + T
7 1−iT 1 - iT
11 1+T 1 + T
13 1 1
19 1+iT 1 + iT
23 1+T 1 + T
29 1−T 1 - T
31 1 1
37 1−iT 1 - iT
41 1−iT 1 - iT
43 1−T 1 - T
47 1+iT 1 + iT
53 1 1
59 1+T 1 + T
61 1 1
67 1−T 1 - T
71 1−T 1 - T
73 1 1
79 1+T 1 + T
83 1−iT 1 - iT
89 1+iT 1 + iT
97 1−iT 1 - iT
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   L(s)=∏p (1−αp p−s)−1L(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 ZZ-function along the critical line