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Properties

Label 2-228-57.2-c1-0-6
Degree $2$
Conductor $228$
Sign $0.209 + 0.977i$
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  + (1.11 − 1.32i)3-s + (−2.64 − 4.58i)7-s + (−0.520 − 2.95i)9-s + (3.74 + 4.46i)13-s + (3.5 + 2.59i)19-s + (−9.02 − 1.59i)21-s + (3.83 − 3.21i)25-s + (−4.5 − 2.59i)27-s + (−1.84 + 1.06i)31-s + 11.0i·37-s + 10.0·39-s + (8.31 − 3.02i)43-s + (−10.4 + 18.1i)49-s + (7.34 − 1.75i)57-s + (−14.5 − 5.30i)61-s + ⋯
L(s)  = 1  + (0.642 − 0.766i)3-s + (−0.999 − 1.73i)7-s + (−0.173 − 0.984i)9-s + (1.03 + 1.23i)13-s + (0.802 + 0.596i)19-s + (−1.96 − 0.347i)21-s + (0.766 − 0.642i)25-s + (−0.866 − 0.499i)27-s + (−0.332 + 0.191i)31-s + 1.81i·37-s + 1.61·39-s + (1.26 − 0.461i)43-s + (−1.49 + 2.59i)49-s + (0.972 − 0.231i)57-s + (−1.86 − 0.679i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 228 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.209 + 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 228 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.209 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(228\)    =    \(2^{2} \cdot 3 \cdot 19\)
Sign: $0.209 + 0.977i$
Analytic conductor: \(1.82058\)
Root analytic conductor: \(1.34929\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{228} (173, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 228,\ (\ :1/2),\ 0.209 + 0.977i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.06652 - 0.862331i\)
\(L(\frac12)\) \(\approx\) \(1.06652 - 0.862331i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.11 + 1.32i)T \)
19 \( 1 + (-3.5 - 2.59i)T \)
good5 \( 1 + (-3.83 + 3.21i)T^{2} \)
7 \( 1 + (2.64 + 4.58i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3.74 - 4.46i)T + (-2.25 + 12.8i)T^{2} \)
17 \( 1 + (15.9 + 5.81i)T^{2} \)
23 \( 1 + (-17.6 - 14.7i)T^{2} \)
29 \( 1 + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (1.84 - 1.06i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 11.0iT - 37T^{2} \)
41 \( 1 + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (-8.31 + 3.02i)T + (32.9 - 27.6i)T^{2} \)
47 \( 1 + (44.1 - 16.0i)T^{2} \)
53 \( 1 + (40.6 + 34.0i)T^{2} \)
59 \( 1 + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (14.5 + 5.30i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (-7.51 + 1.32i)T + (62.9 - 22.9i)T^{2} \)
71 \( 1 + (54.3 - 45.6i)T^{2} \)
73 \( 1 + (-3.56 - 2.99i)T + (12.6 + 71.8i)T^{2} \)
79 \( 1 + (6.87 - 8.19i)T + (-13.7 - 77.7i)T^{2} \)
83 \( 1 + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (15.4 - 87.6i)T^{2} \)
97 \( 1 + (-18.7 - 3.30i)T + (91.1 + 33.1i)T^{2} \)
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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

−12.21346259752382893916208514470, −11.03645327595984713020515778814, −9.982111107502784728769989386653, −9.079986706709276967415379794704, −7.899152485762410496785068987032, −6.92579610946042669814715179356, −6.33001524837412524161229141905, −4.17326586366538287669357329550, −3.24633576169437683731331243258, −1.21778388705932726042727054398, 2.64509883217163595027761310662, 3.48477090891029758783827219722, 5.25654672109020584832808946274, 6.01144648306307085511353478296, 7.67352271563414778587337255567, 8.941537121992945396319009243470, 9.183799436791608740191085571626, 10.39849670699388881221375637057, 11.35780475935485299114323614329, 12.63126959272831607208460618288

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