Properties

Label 1-675-675.229-r0-0-0
Degree 11
Conductor 675675
Sign 0.2800.959i-0.280 - 0.959i
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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.961 − 0.275i)2-s + (0.848 + 0.529i)4-s + (−0.766 − 0.642i)7-s + (−0.669 − 0.743i)8-s + (−0.719 + 0.694i)11-s + (−0.961 + 0.275i)13-s + (0.559 + 0.829i)14-s + (0.438 + 0.898i)16-s + (0.978 − 0.207i)17-s + (0.669 + 0.743i)19-s + (0.882 − 0.469i)22-s + (0.997 − 0.0697i)23-s + 26-s + (−0.309 − 0.951i)28-s + (−0.615 − 0.788i)29-s + ⋯
L(s)  = 1  + (−0.961 − 0.275i)2-s + (0.848 + 0.529i)4-s + (−0.766 − 0.642i)7-s + (−0.669 − 0.743i)8-s + (−0.719 + 0.694i)11-s + (−0.961 + 0.275i)13-s + (0.559 + 0.829i)14-s + (0.438 + 0.898i)16-s + (0.978 − 0.207i)17-s + (0.669 + 0.743i)19-s + (0.882 − 0.469i)22-s + (0.997 − 0.0697i)23-s + 26-s + (−0.309 − 0.951i)28-s + (−0.615 − 0.788i)29-s + ⋯

Functional equation

Λ(s)=(675s/2ΓR(s)L(s)=((0.2800.959i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.280 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(675s/2ΓR(s)L(s)=((0.2800.959i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.280 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 675675    =    33523^{3} \cdot 5^{2}
Sign: 0.2800.959i-0.280 - 0.959i
Analytic conductor: 3.134683.13468
Root analytic conductor: 3.134683.13468
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ675(229,)\chi_{675} (229, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 675, (0: ), 0.2800.959i)(1,\ 675,\ (0:\ ),\ -0.280 - 0.959i)

Particular Values

L(12)L(\frac{1}{2}) \approx 0.30478040800.4064216225i0.3047804080 - 0.4064216225i
L(12)L(\frac12) \approx 0.30478040800.4064216225i0.3047804080 - 0.4064216225i
L(1)L(1) \approx 0.55895142120.1348842514i0.5589514212 - 0.1348842514i
L(1)L(1) \approx 0.55895142120.1348842514i0.5589514212 - 0.1348842514i

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+(0.9610.275i)T 1 + (-0.961 - 0.275i)T
7 1+(0.7660.642i)T 1 + (-0.766 - 0.642i)T
11 1+(0.719+0.694i)T 1 + (-0.719 + 0.694i)T
13 1+(0.961+0.275i)T 1 + (-0.961 + 0.275i)T
17 1+(0.9780.207i)T 1 + (0.978 - 0.207i)T
19 1+(0.669+0.743i)T 1 + (0.669 + 0.743i)T
23 1+(0.9970.0697i)T 1 + (0.997 - 0.0697i)T
29 1+(0.6150.788i)T 1 + (-0.615 - 0.788i)T
31 1+(0.3740.927i)T 1 + (-0.374 - 0.927i)T
37 1+(0.9130.406i)T 1 + (-0.913 - 0.406i)T
41 1+(0.9610.275i)T 1 + (0.961 - 0.275i)T
43 1+(0.173+0.984i)T 1 + (-0.173 + 0.984i)T
47 1+(0.3740.927i)T 1 + (0.374 - 0.927i)T
53 1+(0.3090.951i)T 1 + (-0.309 - 0.951i)T
59 1+(0.7190.694i)T 1 + (-0.719 - 0.694i)T
61 1+(0.2410.970i)T 1 + (-0.241 - 0.970i)T
67 1+(0.6150.788i)T 1 + (0.615 - 0.788i)T
71 1+(0.6690.743i)T 1 + (0.669 - 0.743i)T
73 1+(0.913+0.406i)T 1 + (-0.913 + 0.406i)T
79 1+(0.6150.788i)T 1 + (-0.615 - 0.788i)T
83 1+(0.03480.999i)T 1 + (-0.0348 - 0.999i)T
89 1+(0.9130.406i)T 1 + (0.913 - 0.406i)T
97 1+(0.990+0.139i)T 1 + (-0.990 + 0.139i)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

−23.1395870539330754114481352162, −22.07026915687694178914448654555, −21.31288115392233238973903254369, −20.34816623416795591615839958195, −19.43895404487620687396145308103, −18.910912179173659885126240484449, −18.181221193363929936119071088681, −17.222505526790065663065672645803, −16.44190192716360621308962819806, −15.73003617774106458824709245850, −15.02514419239059381281851886405, −14.04414593658951414106777071216, −12.79980052238835490932683077211, −12.082761819112459741497781979, −10.99440134605465433111908594643, −10.22854967362588214029906961032, −9.334965031081699701160444228864, −8.69591409541768543901153704287, −7.58116724482886400647827798805, −6.93399110069232732572461187513, −5.69137952999830684916177718700, −5.214127468059619194711124421302, −3.17340412484624597051045537621, −2.64367618583787976284536430954, −1.11403418747516941067059202354, 0.377765650094207629114563477398, 1.80743901853855722796635856581, 2.88128394076324538468108571744, 3.79405089784161041426609912171, 5.175383158858211394475977644810, 6.39163067715567231882631618682, 7.52560119295411235321144534284, 7.65647856150428081841012197659, 9.24825771835173357524196274075, 9.80848723385930426491938752241, 10.431426054372766469051299303442, 11.49992254257665131922500468741, 12.43840095022062947887226618512, 13.0312641507818199608823495044, 14.30703253612525164193552043745, 15.283891652996867699434716964078, 16.19243233696779705404743020328, 16.86289209379940141540796372809, 17.509528873996813202787077648007, 18.646133984001188970813807894423, 19.07682356758589390389011404892, 20.05372817050168367116282095457, 20.663036769442183379710241884963, 21.413490743187805864294697657, 22.59929177832188146247307867878

Graph of the ZZ-function along the critical line