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Properties

Label 2-228-19.7-c1-0-3
Degree $2$
Conductor $228$
Sign $-0.856 + 0.516i$
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  + (−0.5 − 0.866i)3-s + (−1.22 − 2.12i)5-s − 3.44·7-s + (−0.499 + 0.866i)9-s − 2.44·11-s + (0.5 − 0.866i)13-s + (−1.22 + 2.12i)15-s + (−2.44 − 4.24i)17-s + (1 + 4.24i)19-s + (1.72 + 2.98i)21-s + (4.22 − 7.31i)23-s + (−0.499 + 0.866i)25-s + 0.999·27-s + (2.44 − 4.24i)29-s − 9.44·31-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (−0.547 − 0.948i)5-s − 1.30·7-s + (−0.166 + 0.288i)9-s − 0.738·11-s + (0.138 − 0.240i)13-s + (−0.316 + 0.547i)15-s + (−0.594 − 1.02i)17-s + (0.229 + 0.973i)19-s + (0.376 + 0.651i)21-s + (0.880 − 1.52i)23-s + (−0.0999 + 0.173i)25-s + 0.192·27-s + (0.454 − 0.787i)29-s − 1.69·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 228 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.856 + 0.516i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 228 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.856 + 0.516i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(228\)    =    \(2^{2} \cdot 3 \cdot 19\)
Sign: $-0.856 + 0.516i$
Analytic conductor: \(1.82058\)
Root analytic conductor: \(1.34929\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{228} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 228,\ (\ :1/2),\ -0.856 + 0.516i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.147989 - 0.532423i\)
\(L(\frac12)\) \(\approx\) \(0.147989 - 0.532423i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (-1 - 4.24i)T \)
good5 \( 1 + (1.22 + 2.12i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + 3.44T + 7T^{2} \)
11 \( 1 + 2.44T + 11T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.44 + 4.24i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-4.22 + 7.31i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.44 + 4.24i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 9.44T + 31T^{2} \)
37 \( 1 - 8.79T + 37T^{2} \)
41 \( 1 + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.72 - 2.98i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.550 - 0.953i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.22 + 2.12i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.67 - 6.36i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.05 + 1.81i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.27 + 3.94i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (3 + 5.19i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-5.94 - 10.3i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.17 + 10.6i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 16.8T + 83T^{2} \)
89 \( 1 + (-1.77 + 3.07i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.34 + 9.26i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.06556665260009257446371428325, −10.96620344738525493467801047297, −9.833242336422328866808178016073, −8.824872867938208424202919150592, −7.82409790010531962056188731439, −6.74921218838841099093251797580, −5.63090179848113766887163790309, −4.39994799330438832216427953702, −2.80636642808607201964967165860, −0.45443199118390126378249780430, 2.93335150728649033945334877980, 3.84199354696670198863612260993, 5.43746574873831591168911730382, 6.63149580114064295325483813593, 7.38825129875892642756158410757, 8.920898863990766317912420989497, 9.805809131960012047621851744680, 10.86489111296415438464868511636, 11.29534099761567027516651708160, 12.73239152784286069516796074703

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