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Properties

Label 2-228-19.9-c1-0-0
Degree $2$
Conductor $228$
Sign $-0.0562 - 0.998i$
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  + (−0.766 + 0.642i)3-s + (0.434 + 0.158i)5-s + (−2.12 + 3.67i)7-s + (0.173 − 0.984i)9-s + (2.54 + 4.40i)11-s + (−3.32 − 2.79i)13-s + (−0.434 + 0.158i)15-s + (1.12 + 6.36i)17-s + (4.31 + 0.582i)19-s + (−0.736 − 4.17i)21-s + (−1.45 + 0.530i)23-s + (−3.66 − 3.07i)25-s + (0.500 + 0.866i)27-s + (−0.495 + 2.81i)29-s + (2.16 − 3.75i)31-s + ⋯
L(s)  = 1  + (−0.442 + 0.371i)3-s + (0.194 + 0.0707i)5-s + (−0.801 + 1.38i)7-s + (0.0578 − 0.328i)9-s + (0.767 + 1.32i)11-s + (−0.922 − 0.774i)13-s + (−0.112 + 0.0408i)15-s + (0.272 + 1.54i)17-s + (0.991 + 0.133i)19-s + (−0.160 − 0.911i)21-s + (−0.304 + 0.110i)23-s + (−0.733 − 0.615i)25-s + (0.0962 + 0.166i)27-s + (−0.0920 + 0.522i)29-s + (0.389 − 0.674i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 228 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0562 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 228 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0562 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(228\)    =    \(2^{2} \cdot 3 \cdot 19\)
Sign: $-0.0562 - 0.998i$
Analytic conductor: \(1.82058\)
Root analytic conductor: \(1.34929\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{228} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 228,\ (\ :1/2),\ -0.0562 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.644429 + 0.681733i\)
\(L(\frac12)\) \(\approx\) \(0.644429 + 0.681733i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.766 - 0.642i)T \)
19 \( 1 + (-4.31 - 0.582i)T \)
good5 \( 1 + (-0.434 - 0.158i)T + (3.83 + 3.21i)T^{2} \)
7 \( 1 + (2.12 - 3.67i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.54 - 4.40i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.32 + 2.79i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (-1.12 - 6.36i)T + (-15.9 + 5.81i)T^{2} \)
23 \( 1 + (1.45 - 0.530i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (0.495 - 2.81i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (-2.16 + 3.75i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 3.48T + 37T^{2} \)
41 \( 1 + (-8.79 + 7.38i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (5.07 + 1.84i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (-1.86 + 10.5i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (-8.22 + 2.99i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (-2.14 - 12.1i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-3.32 + 1.21i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (0.752 - 4.26i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (5.89 + 2.14i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (0.545 - 0.458i)T + (12.6 - 71.8i)T^{2} \)
79 \( 1 + (5.21 - 4.37i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (-0.390 + 0.675i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-1.56 - 1.31i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (0.104 + 0.594i)T + (-91.1 + 33.1i)T^{2} \)
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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

−12.20560791338646521422010411109, −11.90596928681410617272834246022, −10.19195773231325365278641782628, −9.800493480995298606842284977134, −8.770095517804136253840763848884, −7.37169111505660352364627790090, −6.12047382375364826343536300354, −5.39427100140721612453571451956, −3.90153711537805315508191602774, −2.30530166902156775601991626276, 0.847273641629958406918971337733, 3.14605483390952207479617959225, 4.51873200128923544583342028315, 5.94824262825145657656395682309, 6.93175183288794701167562972150, 7.67308536205677618132761624765, 9.340665579034680155156786362507, 9.906281724894661124879697146105, 11.26017856682375527298108377186, 11.76810442278251368301598556977

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