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Properties

Label 1-15e2-225.142-r1-0-0
Degree $1$
Conductor $225$
Sign $0.663 - 0.747i$
Analytic cond. $24.1796$
Root an. cond. $24.1796$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

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

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

Invariants

Degree: \(1\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.663 - 0.747i$
Analytic conductor: \(24.1796\)
Root analytic conductor: \(24.1796\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (142, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 225,\ (1:\ ),\ 0.663 - 0.747i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.034821875 - 0.4650673910i\)
\(L(\frac12)\) \(\approx\) \(1.034821875 - 0.4650673910i\)
\(L(1)\) \(\approx\) \(0.8504837098 + 0.1145423349i\)
\(L(1)\) \(\approx\) \(0.8504837098 + 0.1145423349i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 + (-0.406 + 0.913i)T \)
7 \( 1 + (0.866 - 0.5i)T \)
11 \( 1 + (0.913 + 0.406i)T \)
13 \( 1 + (-0.406 - 0.913i)T \)
17 \( 1 + (-0.951 + 0.309i)T \)
19 \( 1 + (-0.309 - 0.951i)T \)
23 \( 1 + (-0.994 + 0.104i)T \)
29 \( 1 + (0.978 - 0.207i)T \)
31 \( 1 + (-0.978 - 0.207i)T \)
37 \( 1 + (0.587 - 0.809i)T \)
41 \( 1 + (0.913 - 0.406i)T \)
43 \( 1 + (-0.866 + 0.5i)T \)
47 \( 1 + (-0.207 - 0.978i)T \)
53 \( 1 + (-0.951 - 0.309i)T \)
59 \( 1 + (-0.913 + 0.406i)T \)
61 \( 1 + (0.913 + 0.406i)T \)
67 \( 1 + (0.207 - 0.978i)T \)
71 \( 1 + (0.309 - 0.951i)T \)
73 \( 1 + (0.587 + 0.809i)T \)
79 \( 1 + (0.978 - 0.207i)T \)
83 \( 1 + (-0.743 - 0.669i)T \)
89 \( 1 + (0.809 - 0.587i)T \)
97 \( 1 + (-0.207 - 0.978i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

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 $Z$-function along the critical line