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Properties

Label 2-228-76.27-c1-0-9
Degree $2$
Conductor $228$
Sign $-0.223 + 0.974i$
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  + (−0.524 − 1.31i)2-s + (−0.5 + 0.866i)3-s + (−1.44 + 1.37i)4-s + (−0.490 + 0.849i)5-s + (1.39 + 0.202i)6-s − 4.28i·7-s + (2.57 + 1.17i)8-s + (−0.499 − 0.866i)9-s + (1.37 + 0.198i)10-s − 3.77i·11-s + (−0.469 − 1.94i)12-s + (4.37 − 2.52i)13-s + (−5.62 + 2.24i)14-s + (−0.490 − 0.849i)15-s + (0.199 − 3.99i)16-s + (0.354 − 0.613i)17-s + ⋯
L(s)  = 1  + (−0.371 − 0.928i)2-s + (−0.288 + 0.499i)3-s + (−0.724 + 0.689i)4-s + (−0.219 + 0.379i)5-s + (0.571 + 0.0824i)6-s − 1.61i·7-s + (0.908 + 0.416i)8-s + (−0.166 − 0.288i)9-s + (0.433 + 0.0626i)10-s − 1.13i·11-s + (−0.135 − 0.561i)12-s + (1.21 − 0.701i)13-s + (−1.50 + 0.601i)14-s + (−0.126 − 0.219i)15-s + (0.0498 − 0.998i)16-s + (0.0859 − 0.148i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 228 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.223 + 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 228 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.223 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(228\)    =    \(2^{2} \cdot 3 \cdot 19\)
Sign: $-0.223 + 0.974i$
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: \((2,\ 228,\ (\ :1/2),\ -0.223 + 0.974i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.493568 - 0.619386i\)
\(L(\frac12)\) \(\approx\) \(0.493568 - 0.619386i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.524 + 1.31i)T \)
3 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (-2.31 + 3.69i)T \)
good5 \( 1 + (0.490 - 0.849i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + 4.28iT - 7T^{2} \)
11 \( 1 + 3.77iT - 11T^{2} \)
13 \( 1 + (-4.37 + 2.52i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.354 + 0.613i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (4.89 - 2.82i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.94 - 2.27i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 3.03T + 31T^{2} \)
37 \( 1 + 5.22iT - 37T^{2} \)
41 \( 1 + (-1.10 - 0.638i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.92 - 2.84i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.130 - 0.0751i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-9.82 + 5.67i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.36 + 5.82i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.30 + 10.9i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.08 - 8.79i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (1.60 - 2.77i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (7.73 - 13.3i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.16 + 5.47i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 15.8iT - 83T^{2} \)
89 \( 1 + (-3.25 + 1.87i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.29 - 0.749i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.29279761024026216230355578413, −11.10143020322675750334535801806, −10.31927924695015977068983598128, −9.273370284807520580075999722122, −8.123576359352894488183367729737, −7.13565279461280460606152356647, −5.50266787200429004154643574227, −3.92005677758220584658348314855, −3.36773182429269222742498737956, −0.836641805646777477256593194608, 1.84469056932887198272200286713, 4.33229974916957655472757978175, 5.64235550165530587475094926826, 6.28000621316584785550452532576, 7.56178041182486111997812814195, 8.534633789760651469507071727162, 9.198153189449699975205607028558, 10.37687230689211326073004790292, 11.84558551045951370705307771803, 12.38682695519385807245895455677

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