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Properties

Label 2-228-76.27-c1-0-2
Degree $2$
Conductor $228$
Sign $-0.695 - 0.718i$
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  + (−1.09 + 0.892i)2-s + (−0.5 + 0.866i)3-s + (0.408 − 1.95i)4-s + (0.127 − 0.220i)5-s + (−0.223 − 1.39i)6-s + 1.43i·7-s + (1.29 + 2.51i)8-s + (−0.499 − 0.866i)9-s + (0.0569 + 0.354i)10-s + 6.05i·11-s + (1.49 + 1.33i)12-s + (−0.665 + 0.384i)13-s + (−1.27 − 1.57i)14-s + (0.127 + 0.220i)15-s + (−3.66 − 1.59i)16-s + (−1.84 + 3.19i)17-s + ⋯
L(s)  = 1  + (−0.775 + 0.630i)2-s + (−0.288 + 0.499i)3-s + (0.204 − 0.978i)4-s + (0.0568 − 0.0984i)5-s + (−0.0914 − 0.570i)6-s + 0.542i·7-s + (0.459 + 0.888i)8-s + (−0.166 − 0.288i)9-s + (0.0179 + 0.112i)10-s + 1.82i·11-s + (0.430 + 0.384i)12-s + (−0.184 + 0.106i)13-s + (−0.342 − 0.420i)14-s + (0.0328 + 0.0568i)15-s + (−0.916 − 0.399i)16-s + (−0.447 + 0.775i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 228 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.695 - 0.718i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 228 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.695 - 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(228\)    =    \(2^{2} \cdot 3 \cdot 19\)
Sign: $-0.695 - 0.718i$
Analytic conductor: \(1.82058\)
Root analytic conductor: \(1.34929\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{228} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 228,\ (\ :1/2),\ -0.695 - 0.718i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.255087 + 0.601479i\)
\(L(\frac12)\) \(\approx\) \(0.255087 + 0.601479i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.09 - 0.892i)T \)
3 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (-0.741 + 4.29i)T \)
good5 \( 1 + (-0.127 + 0.220i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 - 1.43iT - 7T^{2} \)
11 \( 1 - 6.05iT - 11T^{2} \)
13 \( 1 + (0.665 - 0.384i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.84 - 3.19i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (7.31 - 4.22i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.66 + 2.69i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 1.97T + 31T^{2} \)
37 \( 1 - 3.75iT - 37T^{2} \)
41 \( 1 + (1.68 + 0.972i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.94 - 2.85i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.39 + 0.808i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-10.8 + 6.28i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.994 + 1.72i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.538 - 0.932i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.68 + 4.64i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-5.28 + 9.15i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (1.44 - 2.50i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.35 + 4.07i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 5.30iT - 83T^{2} \)
89 \( 1 + (11.0 - 6.37i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-11.3 - 6.57i)T + (48.5 + 84.0i)T^{2} \)
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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

−12.36805681236332600483814766411, −11.47423846074673474015552651922, −10.30982562266308651250759648584, −9.639115407839646795102447827956, −8.816399616692609676138383179656, −7.59112516807626829118655778018, −6.60954444855716357760611427683, −5.43279627219841223611245864462, −4.42973442692607569287043150902, −2.06818502426862954239175819200, 0.72016904637395250179581286384, 2.61443711624762308120584302843, 4.01258373785828858242261803425, 5.88237159740815249034494736191, 6.98695475428789824779060145269, 8.089476787083869927213331090591, 8.800745636720974401894703964682, 10.22161067925505517482303518430, 10.79852131259687427057305942459, 11.78444612354485972567087230119

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