Properties

Label 1-405-405.68-r0-0-0
Degree 11
Conductor 405405
Sign 0.860+0.509i-0.860 + 0.509i
Analytic cond. 1.880811.88081
Root an. cond. 1.880811.88081
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  + (−0.802 + 0.597i)2-s + (0.286 − 0.957i)4-s + (0.727 + 0.686i)7-s + (0.342 + 0.939i)8-s + (−0.893 + 0.448i)11-s + (−0.918 + 0.396i)13-s + (−0.993 − 0.116i)14-s + (−0.835 − 0.549i)16-s + (0.642 + 0.766i)17-s + (−0.766 − 0.642i)19-s + (0.448 − 0.893i)22-s + (−0.727 + 0.686i)23-s + (0.5 − 0.866i)26-s + (0.866 − 0.5i)28-s + (−0.993 + 0.116i)29-s + ⋯
L(s)  = 1  + (−0.802 + 0.597i)2-s + (0.286 − 0.957i)4-s + (0.727 + 0.686i)7-s + (0.342 + 0.939i)8-s + (−0.893 + 0.448i)11-s + (−0.918 + 0.396i)13-s + (−0.993 − 0.116i)14-s + (−0.835 − 0.549i)16-s + (0.642 + 0.766i)17-s + (−0.766 − 0.642i)19-s + (0.448 − 0.893i)22-s + (−0.727 + 0.686i)23-s + (0.5 − 0.866i)26-s + (0.866 − 0.5i)28-s + (−0.993 + 0.116i)29-s + ⋯

Functional equation

Λ(s)=(405s/2ΓR(s)L(s)=((0.860+0.509i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.860 + 0.509i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(405s/2ΓR(s)L(s)=((0.860+0.509i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.860 + 0.509i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 405405    =    3453^{4} \cdot 5
Sign: 0.860+0.509i-0.860 + 0.509i
Analytic conductor: 1.880811.88081
Root analytic conductor: 1.880811.88081
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ405(68,)\chi_{405} (68, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 405, (0: ), 0.860+0.509i)(1,\ 405,\ (0:\ ),\ -0.860 + 0.509i)

Particular Values

L(12)L(\frac{1}{2}) \approx 0.1484976303+0.5426975094i0.1484976303 + 0.5426975094i
L(12)L(\frac12) \approx 0.1484976303+0.5426975094i0.1484976303 + 0.5426975094i
L(1)L(1) \approx 0.5677567659+0.3011414098i0.5677567659 + 0.3011414098i
L(1)L(1) \approx 0.5677567659+0.3011414098i0.5677567659 + 0.3011414098i

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad3 1 1
5 1 1
good2 1+(0.802+0.597i)T 1 + (-0.802 + 0.597i)T
7 1+(0.727+0.686i)T 1 + (0.727 + 0.686i)T
11 1+(0.893+0.448i)T 1 + (-0.893 + 0.448i)T
13 1+(0.918+0.396i)T 1 + (-0.918 + 0.396i)T
17 1+(0.642+0.766i)T 1 + (0.642 + 0.766i)T
19 1+(0.7660.642i)T 1 + (-0.766 - 0.642i)T
23 1+(0.727+0.686i)T 1 + (-0.727 + 0.686i)T
29 1+(0.993+0.116i)T 1 + (-0.993 + 0.116i)T
31 1+(0.9730.230i)T 1 + (0.973 - 0.230i)T
37 1+(0.9840.173i)T 1 + (0.984 - 0.173i)T
41 1+(0.597+0.802i)T 1 + (-0.597 + 0.802i)T
43 1+(0.9980.0581i)T 1 + (-0.998 - 0.0581i)T
47 1+(0.230+0.973i)T 1 + (-0.230 + 0.973i)T
53 1+(0.866+0.5i)T 1 + (-0.866 + 0.5i)T
59 1+(0.893+0.448i)T 1 + (0.893 + 0.448i)T
61 1+(0.2860.957i)T 1 + (-0.286 - 0.957i)T
67 1+(0.116+0.993i)T 1 + (-0.116 + 0.993i)T
71 1+(0.939+0.342i)T 1 + (0.939 + 0.342i)T
73 1+(0.3420.939i)T 1 + (-0.342 - 0.939i)T
79 1+(0.5970.802i)T 1 + (-0.597 - 0.802i)T
83 1+(0.8020.597i)T 1 + (0.802 - 0.597i)T
89 1+(0.939+0.342i)T 1 + (-0.939 + 0.342i)T
97 1+(0.549+0.835i)T 1 + (-0.549 + 0.835i)T
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   L(s)=p (1αpps)1L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−24.10361362386469726867411984196, −23.06379923503337875430297139806, −22.03630026376413968011332469806, −21.04067396725087930279056404726, −20.56855009917833284759098330807, −19.66307248411057468687184479795, −18.65135847036034343696313153703, −18.05052674434539483252247424532, −16.98725592184498180738353401389, −16.495255799378405626622985825070, −15.26266047539316017129774841551, −14.1547707264071881205243350670, −13.14737764487833832875308060860, −12.18896432083288000340667955407, −11.29557473149284132886022348583, −10.34941634034489148189035977103, −9.82707564327056906846244887799, −8.31614845971689378627806860679, −7.887865846004559003568367445672, −6.859447022479149982210245778646, −5.30115653866991875812160106804, −4.15777323096005154603077709312, −2.940421396772922472485577914599, −1.879416087574081320121585548926, −0.41179945595950939092545481793, 1.654928351442686683479568920847, 2.54827918026250769883967896993, 4.55815583134634373336947563088, 5.391993215309500032555830505976, 6.398417439490760220890385700718, 7.658139532819536058373119008638, 8.14759091009233406683114242085, 9.3128188556228152269406910073, 10.09750204047206017881294144929, 11.12720872823633736109272082387, 12.04256185447591521949378026046, 13.241684984872373700411693468148, 14.58872972071446141362386368944, 15.01875401976241364286493339477, 15.90475972696474286724394394498, 17.023822203890635715488533182, 17.63399457179764130898758075124, 18.5291377433820512716017741017, 19.23142427583638895925347252082, 20.20579325378811297786367478773, 21.203808510254329983474805763017, 22.013536690352963290778712735998, 23.44894918092018187241120702452, 23.88436008326287100807279539310, 24.77259961284391456819738518933

Graph of the ZZ-function along the critical line