Properties

Label 1-15e2-225.142-r1-0-0
Degree 11
Conductor 225225
Sign 0.6630.747i0.663 - 0.747i
Analytic cond. 24.179624.1796
Root an. cond. 24.179624.1796
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  + (−0.406 + 0.913i)2-s + (−0.669 − 0.743i)4-s + (0.866 − 0.5i)7-s + (0.951 − 0.309i)8-s + (0.913 + 0.406i)11-s + (−0.406 − 0.913i)13-s + (0.104 + 0.994i)14-s + (−0.104 + 0.994i)16-s + (−0.951 + 0.309i)17-s + (−0.309 − 0.951i)19-s + (−0.743 + 0.669i)22-s + (−0.994 + 0.104i)23-s + 26-s + (−0.951 − 0.309i)28-s + (0.978 − 0.207i)29-s + ⋯
L(s)  = 1  + (−0.406 + 0.913i)2-s + (−0.669 − 0.743i)4-s + (0.866 − 0.5i)7-s + (0.951 − 0.309i)8-s + (0.913 + 0.406i)11-s + (−0.406 − 0.913i)13-s + (0.104 + 0.994i)14-s + (−0.104 + 0.994i)16-s + (−0.951 + 0.309i)17-s + (−0.309 − 0.951i)19-s + (−0.743 + 0.669i)22-s + (−0.994 + 0.104i)23-s + 26-s + (−0.951 − 0.309i)28-s + (0.978 − 0.207i)29-s + ⋯

Functional equation

Λ(s)=(225s/2ΓR(s+1)L(s)=((0.6630.747i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.663 - 0.747i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(225s/2ΓR(s+1)L(s)=((0.6630.747i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.663 - 0.747i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 225225    =    32523^{2} \cdot 5^{2}
Sign: 0.6630.747i0.663 - 0.747i
Analytic conductor: 24.179624.1796
Root analytic conductor: 24.179624.1796
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ225(142,)\chi_{225} (142, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 225, (1: ), 0.6630.747i)(1,\ 225,\ (1:\ ),\ 0.663 - 0.747i)

Particular Values

L(12)L(\frac{1}{2}) \approx 1.0348218750.4650673910i1.034821875 - 0.4650673910i
L(12)L(\frac12) \approx 1.0348218750.4650673910i1.034821875 - 0.4650673910i
L(1)L(1) \approx 0.8504837098+0.1145423349i0.8504837098 + 0.1145423349i
L(1)L(1) \approx 0.8504837098+0.1145423349i0.8504837098 + 0.1145423349i

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+(0.406+0.913i)T 1 + (-0.406 + 0.913i)T
7 1+(0.8660.5i)T 1 + (0.866 - 0.5i)T
11 1+(0.913+0.406i)T 1 + (0.913 + 0.406i)T
13 1+(0.4060.913i)T 1 + (-0.406 - 0.913i)T
17 1+(0.951+0.309i)T 1 + (-0.951 + 0.309i)T
19 1+(0.3090.951i)T 1 + (-0.309 - 0.951i)T
23 1+(0.994+0.104i)T 1 + (-0.994 + 0.104i)T
29 1+(0.9780.207i)T 1 + (0.978 - 0.207i)T
31 1+(0.9780.207i)T 1 + (-0.978 - 0.207i)T
37 1+(0.5870.809i)T 1 + (0.587 - 0.809i)T
41 1+(0.9130.406i)T 1 + (0.913 - 0.406i)T
43 1+(0.866+0.5i)T 1 + (-0.866 + 0.5i)T
47 1+(0.2070.978i)T 1 + (-0.207 - 0.978i)T
53 1+(0.9510.309i)T 1 + (-0.951 - 0.309i)T
59 1+(0.913+0.406i)T 1 + (-0.913 + 0.406i)T
61 1+(0.913+0.406i)T 1 + (0.913 + 0.406i)T
67 1+(0.2070.978i)T 1 + (0.207 - 0.978i)T
71 1+(0.3090.951i)T 1 + (0.309 - 0.951i)T
73 1+(0.587+0.809i)T 1 + (0.587 + 0.809i)T
79 1+(0.9780.207i)T 1 + (0.978 - 0.207i)T
83 1+(0.7430.669i)T 1 + (-0.743 - 0.669i)T
89 1+(0.8090.587i)T 1 + (0.809 - 0.587i)T
97 1+(0.2070.978i)T 1 + (-0.207 - 0.978i)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

−26.7665856670483010896240402073, −25.482342147762743772365619194559, −24.54679031446176032109844645585, −23.500528707698973947141159658338, −22.0778134132227559936528525909, −21.75737379589625788652291729718, −20.655179297211885429191177220773, −19.78316832834842964734073202562, −18.84303801964349970552666949144, −18.016179213422404793290389028632, −17.11288587233936586538361840335, −16.1436221995463493452145687694, −14.53807227226687646167457402935, −13.92056837738616302391596784222, −12.48851545280139254412516869479, −11.70376136653097890726975304694, −10.98674805496008100956219197314, −9.66806074778819810957591653418, −8.80061780635607203910250765654, −7.91753520986959132403514673094, −6.43940277059398070582110099253, −4.82804118111253549739943806952, −3.89910494184633858581888173923, −2.3620396002793992578402198503, −1.38910687953855340356959438536, 0.45203202733925016807946534625, 1.910656669816904722079151298626, 4.0847019026501770664409082102, 4.95830533543306343270708693171, 6.271409576259364786249997886, 7.28537421126516292618731398293, 8.20012794699823255582506761208, 9.22478934725356535074884082227, 10.33936639790869829252599790042, 11.31779833208518226732273975074, 12.81011057356473898429584693875, 13.92464445174741852914269117340, 14.76269294691061953192599306314, 15.53136578063990939932531006322, 16.75994520850473732550392115043, 17.641104829730485752305693101015, 18.01866571888775662247367174706, 19.70998497490319326695987828371, 19.99204662244164352324611256360, 21.653137509023075548647021975243, 22.53226581383766858907726262359, 23.54940213539606868201842633347, 24.35815944315603882563513026397, 25.05858495085864845339610289099, 26.1000804579136220995965624056

Graph of the ZZ-function along the critical line