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Properties

Label 1-675-675.317-r0-0-0
Degree 11
Conductor 675675
Sign −0.722+0.691i-0.722 + 0.691i
Analytic cond. 3.134683.13468
Root an. cond. 3.134683.13468
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.970 + 0.241i)2-s + (0.882 + 0.469i)4-s + (−0.984 + 0.173i)7-s + (0.743 + 0.669i)8-s + (−0.961 + 0.275i)11-s + (−0.970 + 0.241i)13-s + (−0.997 − 0.0697i)14-s + (0.559 + 0.829i)16-s + (−0.207 + 0.978i)17-s + (−0.669 + 0.743i)19-s + (−0.999 + 0.0348i)22-s + (−0.898 + 0.438i)23-s − 26-s + (−0.951 − 0.309i)28-s + (−0.374 − 0.927i)29-s + ⋯
L(s)  = 1  + (0.970 + 0.241i)2-s + (0.882 + 0.469i)4-s + (−0.984 + 0.173i)7-s + (0.743 + 0.669i)8-s + (−0.961 + 0.275i)11-s + (−0.970 + 0.241i)13-s + (−0.997 − 0.0697i)14-s + (0.559 + 0.829i)16-s + (−0.207 + 0.978i)17-s + (−0.669 + 0.743i)19-s + (−0.999 + 0.0348i)22-s + (−0.898 + 0.438i)23-s − 26-s + (−0.951 − 0.309i)28-s + (−0.374 − 0.927i)29-s + ⋯

Functional equation

Λ(s)=(675s/2 ΓR(s) L(s)=((−0.722+0.691i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.722 + 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(675s/2 ΓR(s) L(s)=((−0.722+0.691i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.722 + 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 675675    =    33⋅523^{3} \cdot 5^{2}
Sign: −0.722+0.691i-0.722 + 0.691i
Analytic conductor: 3.134683.13468
Root analytic conductor: 3.134683.13468
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ675(317,⋅)\chi_{675} (317, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 675, (0: ), −0.722+0.691i)(1,\ 675,\ (0:\ ),\ -0.722 + 0.691i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.5464702989+1.361676130i0.5464702989 + 1.361676130i
L(12)L(\frac12) ≈\approx 0.5464702989+1.361676130i0.5464702989 + 1.361676130i
L(1)L(1) ≈\approx 1.251395382+0.5711708183i1.251395382 + 0.5711708183i
L(1)L(1) ≈\approx 1.251395382+0.5711708183i1.251395382 + 0.5711708183i

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad3 1 1
5 1 1
good2 1+(0.970+0.241i)T 1 + (0.970 + 0.241i)T
7 1+(−0.984+0.173i)T 1 + (-0.984 + 0.173i)T
11 1+(−0.961+0.275i)T 1 + (-0.961 + 0.275i)T
13 1+(−0.970+0.241i)T 1 + (-0.970 + 0.241i)T
17 1+(−0.207+0.978i)T 1 + (-0.207 + 0.978i)T
19 1+(−0.669+0.743i)T 1 + (-0.669 + 0.743i)T
23 1+(−0.898+0.438i)T 1 + (-0.898 + 0.438i)T
29 1+(−0.374−0.927i)T 1 + (-0.374 - 0.927i)T
31 1+(0.990−0.139i)T 1 + (0.990 - 0.139i)T
37 1+(0.406+0.913i)T 1 + (0.406 + 0.913i)T
41 1+(0.241+0.970i)T 1 + (0.241 + 0.970i)T
43 1+(−0.342+0.939i)T 1 + (-0.342 + 0.939i)T
47 1+(0.139−0.990i)T 1 + (0.139 - 0.990i)T
53 1+(0.951+0.309i)T 1 + (0.951 + 0.309i)T
59 1+(0.961+0.275i)T 1 + (0.961 + 0.275i)T
61 1+(−0.719−0.694i)T 1 + (-0.719 - 0.694i)T
67 1+(−0.927−0.374i)T 1 + (-0.927 - 0.374i)T
71 1+(−0.669−0.743i)T 1 + (-0.669 - 0.743i)T
73 1+(0.406−0.913i)T 1 + (0.406 - 0.913i)T
79 1+(0.374+0.927i)T 1 + (0.374 + 0.927i)T
83 1+(−0.529+0.848i)T 1 + (-0.529 + 0.848i)T
89 1+(0.913+0.406i)T 1 + (0.913 + 0.406i)T
97 1+(−0.788−0.615i)T 1 + (-0.788 - 0.615i)T
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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

−22.383594546879477502881662132188, −21.83418103741428524407407111974, −20.86754674970531232841333065964, −20.07889006059141303349524208655, −19.41924846337659472768866794817, −18.599792929017065771990285614881, −17.44108534030414698642343818291, −16.26444614330283385339287190120, −15.8693202539693929986243928700, −14.930613931641112999107819808026, −13.98207117624367109128706744717, −13.21687450001509483098996775695, −12.57994185030767787195882574061, −11.75544144984769800296033110648, −10.61002473918668630640742698111, −10.1025845788057748475762356181, −8.98209835692785271638847822471, −7.532952597689970363434370071795, −6.8804748796235188800809755085, −5.84625680781685076908014771477, −5.00099971146622154992073651150, −4.04241635546561453851593396017, −2.86218680435264850441461634521, −2.35898170635148552874915683294, −0.45324800381059454576507341312, 1.99001563688579297710054512459, 2.790464066922587179117882382050, 3.880825322855144118215560401742, 4.741986061157230711632269281734, 5.88881866677512721187814805685, 6.43826866124459235330689726187, 7.57614396904940138973443647821, 8.29048441053529879739982414515, 9.82023926032497469903406437052, 10.36783937312279993043664804598, 11.65164191481580564477662490731, 12.40796917831689246854845372535, 13.081361826716911846852767990084, 13.782872396361808530692083962373, 15.058664220293135078281078753714, 15.29190760676028086505009339029, 16.452453397013048407042227653000, 16.94186011893772698626871224729, 18.10409854390046264257921600890, 19.29405269546056591308666076400, 19.7774614399894180686634568358, 20.87877808065230810996569485417, 21.56642623093786120015296552717, 22.28198363602482961094526319634, 23.08218708389544559567858673197

Graph of the ZZ-function along the critical line