L-function 1-15e2-225.142-r1-0-0

• 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: $0$

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 + ⋯

Functional equation

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

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

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

Euler product

   $L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$

	$p$	$F_p(T)$
bad	3	$1$
	5	$1$
good	2	$1 + (-0.406 + 0.913i)T$
	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	$1 + (-0.978 - 0.207i)T$

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