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Properties

Label 2-228-57.50-c1-0-3
Degree $2$
Conductor $228$
Sign $0.874 + 0.485i$
Analytic cond. $1.82058$
Root an. cond. $1.34929$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.5 − 0.866i)3-s + 7-s + (1.5 − 2.59i)9-s + (1.5 + 0.866i)13-s + (−4 − 1.73i)19-s + (1.5 − 0.866i)21-s + (−2.5 + 4.33i)25-s − 5.19i·27-s + 1.73i·31-s + 12.1i·37-s + 3·39-s + (−2.5 − 4.33i)43-s − 6·49-s + (−7.5 + 0.866i)57-s + (−6.5 + 11.2i)61-s + ⋯
L(s)  = 1  + (0.866 − 0.499i)3-s + 0.377·7-s + (0.5 − 0.866i)9-s + (0.416 + 0.240i)13-s + (−0.917 − 0.397i)19-s + (0.327 − 0.188i)21-s + (−0.5 + 0.866i)25-s − 0.999i·27-s + 0.311i·31-s + 1.99i·37-s + 0.480·39-s + (−0.381 − 0.660i)43-s − 0.857·49-s + (−0.993 + 0.114i)57-s + (−0.832 + 1.44i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 228 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.874 + 0.485i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 228 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.874 + 0.485i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(228\)    =    \(2^{2} \cdot 3 \cdot 19\)
Sign: $0.874 + 0.485i$
Analytic conductor: \(1.82058\)
Root analytic conductor: \(1.34929\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{228} (221, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 228,\ (\ :1/2),\ 0.874 + 0.485i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.57149 - 0.406737i\)
\(L(\frac12)\) \(\approx\) \(1.57149 - 0.406737i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-1.5 + 0.866i)T \)
19 \( 1 + (4 + 1.73i)T \)
good5 \( 1 + (2.5 - 4.33i)T^{2} \)
7 \( 1 - T + 7T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + (-1.5 - 0.866i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 1.73iT - 31T^{2} \)
37 \( 1 - 12.1iT - 37T^{2} \)
41 \( 1 + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.5 + 4.33i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.5 - 11.2i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (13.5 + 7.79i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-3.5 - 6.06i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-10.5 + 6.06i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-12 + 6.92i)T + (48.5 - 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.23638450314121940984167854519, −11.28705381614395590610145783841, −10.10334026523815425026794321437, −8.998743810241204907394289496518, −8.266788899187782815229867980545, −7.22889587831020399229962245989, −6.19946863192732765769003385777, −4.57303100440988368096496386323, −3.22975089018158342213535268300, −1.71315441715455164188926613272, 2.12575594447982435587548318099, 3.63840437472032277296755898012, 4.70524711034255689501026887986, 6.12261545931633266029500774642, 7.61331070504879108861607496322, 8.380298641344501268660939197580, 9.317552449368671244489144327988, 10.34699076129556622566584011198, 11.11500314513198307906942744420, 12.42835356585981145390035250915

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