↑

Properties

Label 1-30e2-900.419-r0-0-0
Degree $1$
Conductor $900$
Sign $-0.180 - 0.983i$
Analytic cond. $4.17958$
Root an. cond. $4.17958$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)7-s + (−0.104 − 0.994i)11-s + (0.104 − 0.994i)13-s + (0.309 + 0.951i)17-s + (−0.309 − 0.951i)19-s + (−0.913 − 0.406i)23-s + (−0.669 − 0.743i)29-s + (−0.669 + 0.743i)31-s + (0.809 + 0.587i)37-s + (0.104 − 0.994i)41-s + (−0.5 + 0.866i)43-s + (−0.669 − 0.743i)47-s + (−0.5 − 0.866i)49-s + (0.309 − 0.951i)53-s + (−0.104 + 0.994i)59-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)7-s + (−0.104 − 0.994i)11-s + (0.104 − 0.994i)13-s + (0.309 + 0.951i)17-s + (−0.309 − 0.951i)19-s + (−0.913 − 0.406i)23-s + (−0.669 − 0.743i)29-s + (−0.669 + 0.743i)31-s + (0.809 + 0.587i)37-s + (0.104 − 0.994i)41-s + (−0.5 + 0.866i)43-s + (−0.669 − 0.743i)47-s + (−0.5 − 0.866i)49-s + (0.309 − 0.951i)53-s + (−0.104 + 0.994i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.180 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.180 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.180 - 0.983i$
Analytic conductor: \(4.17958\)
Root analytic conductor: \(4.17958\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (419, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 900,\ (0:\ ),\ -0.180 - 0.983i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5308436180 - 0.6371382994i\)
\(L(\frac12)\) \(\approx\) \(0.5308436180 - 0.6371382994i\)
\(L(1)\) \(\approx\) \(0.8616550627 - 0.1242588441i\)
\(L(1)\) \(\approx\) \(0.8616550627 - 0.1242588441i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (-0.104 - 0.994i)T \)
13 \( 1 + (0.104 - 0.994i)T \)
17 \( 1 + (0.309 + 0.951i)T \)
19 \( 1 + (-0.309 - 0.951i)T \)
23 \( 1 + (-0.913 - 0.406i)T \)
29 \( 1 + (-0.669 - 0.743i)T \)
31 \( 1 + (-0.669 + 0.743i)T \)
37 \( 1 + (0.809 + 0.587i)T \)
41 \( 1 + (0.104 - 0.994i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + (-0.669 - 0.743i)T \)
53 \( 1 + (0.309 - 0.951i)T \)
59 \( 1 + (-0.104 + 0.994i)T \)
61 \( 1 + (-0.104 - 0.994i)T \)
67 \( 1 + (0.669 - 0.743i)T \)
71 \( 1 + (0.309 - 0.951i)T \)
73 \( 1 + (0.809 - 0.587i)T \)
79 \( 1 + (-0.669 - 0.743i)T \)
83 \( 1 + (0.978 + 0.207i)T \)
89 \( 1 + (0.809 - 0.587i)T \)
97 \( 1 + (-0.669 - 0.743i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−22.22849457645403087543556295541, −21.34684226081950698335732243167, −20.32847698241367575937288027289, −20.08420562473958128331209571360, −18.9068310794259545480691662833, −18.31121137707819995547061930237, −17.323567514586425657039325324926, −16.49088500491042403631577703916, −16.05257613131992057337950440962, −14.80996808967794599673307080189, −14.193756942217360761033128299705, −13.32596917629353564135170609310, −12.525312926170686851361600553782, −11.67133008741687808059084429697, −10.75977125773912891223298380660, −9.74191548811406497311694138381, −9.4086691160449307184564719046, −7.9769827679160720064433622695, −7.26913122277110406731831825845, −6.52020806444040569839032817046, −5.43289127960866955621089535837, −4.29023197882619455792156098406, −3.71095343353009166184710665371, −2.3704354553127790809084914638, −1.3422336207946266021281127838, 0.36577924355607785431640651524, 1.94257678715413314553307450244, 2.98064104729667442983684760615, 3.73271926674058071501411646443, 5.104110927774722216729751215349, 5.900386232908613437903330840250, 6.53268164080269403065293583295, 7.92704368066602236264885165526, 8.491601955127747656340823229709, 9.404565722651897305400653228, 10.35671986524363694601801529183, 11.13645144922205899550288545451, 12.089784133461308923250502051930, 12.90834484534902928377863609413, 13.50227389842790151734841013812, 14.69870159325634557019156695168, 15.32094058717087228292596323463, 16.11627200236557714160077675607, 16.87112712682822551521476641444, 17.9045824200089909435176948281, 18.534060707441139244352521360611, 19.40497453187905296655264624584, 19.9602387213118479347061566601, 21.11454751845626330702374931499, 21.77413599538148383977207438443

Graph of the $Z$-function along the critical line