Properties

Label 1-475-475.111-r0-0-0
Degree 11
Conductor 475475
Sign 0.560+0.828i-0.560 + 0.828i
Analytic cond. 2.205892.20589
Root an. cond. 2.205892.20589
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  + (0.990 + 0.139i)2-s + (−0.997 + 0.0697i)3-s + (0.961 + 0.275i)4-s + (−0.997 − 0.0697i)6-s + (−0.5 + 0.866i)7-s + (0.913 + 0.406i)8-s + (0.990 − 0.139i)9-s + (−0.978 + 0.207i)11-s + (−0.978 − 0.207i)12-s + (−0.374 + 0.927i)13-s + (−0.615 + 0.788i)14-s + (0.848 + 0.529i)16-s + (−0.719 − 0.694i)17-s + 18-s + (0.438 − 0.898i)21-s + (−0.997 + 0.0697i)22-s + ⋯
L(s)  = 1  + (0.990 + 0.139i)2-s + (−0.997 + 0.0697i)3-s + (0.961 + 0.275i)4-s + (−0.997 − 0.0697i)6-s + (−0.5 + 0.866i)7-s + (0.913 + 0.406i)8-s + (0.990 − 0.139i)9-s + (−0.978 + 0.207i)11-s + (−0.978 − 0.207i)12-s + (−0.374 + 0.927i)13-s + (−0.615 + 0.788i)14-s + (0.848 + 0.529i)16-s + (−0.719 − 0.694i)17-s + 18-s + (0.438 − 0.898i)21-s + (−0.997 + 0.0697i)22-s + ⋯

Functional equation

Λ(s)=(475s/2ΓR(s)L(s)=((0.560+0.828i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.560 + 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(475s/2ΓR(s)L(s)=((0.560+0.828i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.560 + 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 475475    =    52195^{2} \cdot 19
Sign: 0.560+0.828i-0.560 + 0.828i
Analytic conductor: 2.205892.20589
Root analytic conductor: 2.205892.20589
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ475(111,)\chi_{475} (111, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 475, (0: ), 0.560+0.828i)(1,\ 475,\ (0:\ ),\ -0.560 + 0.828i)

Particular Values

L(12)L(\frac{1}{2}) \approx 0.5872289386+1.106004444i0.5872289386 + 1.106004444i
L(12)L(\frac12) \approx 0.5872289386+1.106004444i0.5872289386 + 1.106004444i
L(1)L(1) \approx 1.083743782+0.4757277348i1.083743782 + 0.4757277348i
L(1)L(1) \approx 1.083743782+0.4757277348i1.083743782 + 0.4757277348i

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad5 1 1
19 1 1
good2 1+(0.990+0.139i)T 1 + (0.990 + 0.139i)T
3 1+(0.997+0.0697i)T 1 + (-0.997 + 0.0697i)T
7 1+(0.5+0.866i)T 1 + (-0.5 + 0.866i)T
11 1+(0.978+0.207i)T 1 + (-0.978 + 0.207i)T
13 1+(0.374+0.927i)T 1 + (-0.374 + 0.927i)T
17 1+(0.7190.694i)T 1 + (-0.719 - 0.694i)T
23 1+(0.0348+0.999i)T 1 + (0.0348 + 0.999i)T
29 1+(0.719+0.694i)T 1 + (-0.719 + 0.694i)T
31 1+(0.1040.994i)T 1 + (-0.104 - 0.994i)T
37 1+(0.309+0.951i)T 1 + (0.309 + 0.951i)T
41 1+(0.848+0.529i)T 1 + (0.848 + 0.529i)T
43 1+(0.9390.342i)T 1 + (-0.939 - 0.342i)T
47 1+(0.719+0.694i)T 1 + (-0.719 + 0.694i)T
53 1+(0.961+0.275i)T 1 + (0.961 + 0.275i)T
59 1+(0.882+0.469i)T 1 + (-0.882 + 0.469i)T
61 1+(0.0348+0.999i)T 1 + (0.0348 + 0.999i)T
67 1+(0.438+0.898i)T 1 + (0.438 + 0.898i)T
71 1+(0.559+0.829i)T 1 + (0.559 + 0.829i)T
73 1+(0.3740.927i)T 1 + (-0.374 - 0.927i)T
79 1+(0.997+0.0697i)T 1 + (-0.997 + 0.0697i)T
83 1+(0.1040.994i)T 1 + (-0.104 - 0.994i)T
89 1+(0.8480.529i)T 1 + (0.848 - 0.529i)T
97 1+(0.4380.898i)T 1 + (0.438 - 0.898i)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

−23.10252430632814686198872780277, −22.99829128953046432464352297037, −21.9694840483993133480700097751, −21.23575873340530894121060442529, −20.25144359050944439519949825071, −19.488906880824238819379515880213, −18.36952328519543144908236253725, −17.34257889282611868796227028316, −16.50714887366362814906307078334, −15.78986277867430611856688545809, −14.955265666861824818276546479723, −13.694056110695864399377790173008, −12.86895283537075255129285878287, −12.52559222897192085388379557928, −11.1301165414991474526505585053, −10.63043791191978410815974322068, −9.92275533042105253557260456419, −7.96821384022877678472524876942, −7.05340412693688710928545430616, −6.2170292651751358038797026270, −5.31968445658655096539427994036, −4.44760918316308838624094448879, −3.41980682152757144201135261609, −2.11203758073882374793944633501, −0.52473755973612859345777161286, 1.852103563488171271116064563299, 2.91580596772945985757688816701, 4.28987115396611531639884032313, 5.13128065864817964629668075474, 5.87502311870162551171715748382, 6.79654862305847374044937748582, 7.616951489066477467969719993177, 9.22522820818433543672119852767, 10.22366795916987360466953074922, 11.4395466393824993808911812118, 11.77211057084563925545422215588, 12.91133194667832780109507909259, 13.35000749754730115969542584667, 14.78471908118875242699740152043, 15.57883851668829969596124470804, 16.17015628788471303242516912041, 16.99009920407620172506931315236, 18.12730322259652654682276054963, 18.89380420595908462700577519663, 20.065428197156791628247660023057, 21.16614323320456450810528781458, 21.742839789115570586120676797331, 22.42850588910634229657160324851, 23.173555755852406176952369427821, 24.00557708612125526483090085923

Graph of the ZZ-function along the critical line