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Properties

Label 2-228-57.23-c2-0-2
Degree $2$
Conductor $228$
Sign $-0.0800 - 0.996i$
Analytic cond. $6.21255$
Root an. cond. $2.49249$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.520 − 2.95i)3-s + (−6.99 + 12.1i)7-s + (−8.45 + 3.07i)9-s + (−1.37 + 7.82i)13-s + (5.5 + 18.1i)19-s + (39.4 + 14.3i)21-s + (4.34 − 24.6i)25-s + (13.5 + 23.3i)27-s + (−28.7 + 49.7i)31-s − 48.1·37-s + 23.8·39-s + (−5.94 + 4.98i)43-s + (−73.4 − 127. i)49-s + (50.8 − 25.7i)57-s + (91.1 + 76.5i)61-s + ⋯
L(s)  = 1  + (−0.173 − 0.984i)3-s + (−0.999 + 1.73i)7-s + (−0.939 + 0.342i)9-s + (−0.106 + 0.601i)13-s + (0.289 + 0.957i)19-s + (1.87 + 0.683i)21-s + (0.173 − 0.984i)25-s + (0.5 + 0.866i)27-s + (−0.926 + 1.60i)31-s − 1.30·37-s + 0.611·39-s + (−0.138 + 0.116i)43-s + (−1.49 − 2.59i)49-s + (0.892 − 0.451i)57-s + (1.49 + 1.25i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 228 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0800 - 0.996i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 228 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0800 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(228\)    =    \(2^{2} \cdot 3 \cdot 19\)
Sign: $-0.0800 - 0.996i$
Analytic conductor: \(6.21255\)
Root analytic conductor: \(2.49249\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{228} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 228,\ (\ :1),\ -0.0800 - 0.996i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.489746 + 0.530672i\)
\(L(\frac12)\) \(\approx\) \(0.489746 + 0.530672i\)
\(L(2)\) 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 \( 1 + (0.520 + 2.95i)T \)
19 \( 1 + (-5.5 - 18.1i)T \)
good5 \( 1 + (-4.34 + 24.6i)T^{2} \)
7 \( 1 + (6.99 - 12.1i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + (60.5 - 104. i)T^{2} \)
13 \( 1 + (1.37 - 7.82i)T + (-158. - 57.8i)T^{2} \)
17 \( 1 + (-221. - 185. i)T^{2} \)
23 \( 1 + (-91.8 - 520. i)T^{2} \)
29 \( 1 + (-644. + 540. i)T^{2} \)
31 \( 1 + (28.7 - 49.7i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + 48.1T + 1.36e3T^{2} \)
41 \( 1 + (1.57e3 - 574. i)T^{2} \)
43 \( 1 + (5.94 - 4.98i)T + (321. - 1.82e3i)T^{2} \)
47 \( 1 + (-1.69e3 + 1.41e3i)T^{2} \)
53 \( 1 + (-487. - 2.76e3i)T^{2} \)
59 \( 1 + (-2.66e3 - 2.23e3i)T^{2} \)
61 \( 1 + (-91.1 - 76.5i)T + (646. + 3.66e3i)T^{2} \)
67 \( 1 + (71.1 - 25.9i)T + (3.43e3 - 2.88e3i)T^{2} \)
71 \( 1 + (-875. + 4.96e3i)T^{2} \)
73 \( 1 + (21.5 + 122. i)T + (-5.00e3 + 1.82e3i)T^{2} \)
79 \( 1 + (-7.56 - 42.9i)T + (-5.86e3 + 2.13e3i)T^{2} \)
83 \( 1 + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + (7.44e3 + 2.70e3i)T^{2} \)
97 \( 1 + (-158. - 57.8i)T + (7.20e3 + 6.04e3i)T^{2} \)
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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

−12.23144753329089080312260026751, −11.75801481827070985104308631903, −10.30888417990896390715377618067, −9.072823293918536024800743248103, −8.441430765007337712284955475255, −7.03401905151137776398690518250, −6.16304395490601274805632594830, −5.31508227397593987491625252328, −3.18253695442502300306903304073, −1.96578173509086436091634627663, 0.38364391485338218453710641172, 3.22162436790220638549499573333, 4.07629937881692147862656125440, 5.33299357992221602942877038532, 6.67081121668317806348474407308, 7.63504629277605336980895949371, 9.136195153202038325953204225566, 9.915583822002536593313454978027, 10.65145634409160284973317589855, 11.41360937142323237047172366771

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