Properties

Label 2-228-57.35-c0-0-0
Degree $2$
Conductor $228$
Sign $0.756 - 0.654i$
Analytic cond. $0.113786$
Root an. cond. $0.337323$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.939 + 0.342i)3-s + (0.939 + 1.62i)7-s + (0.766 − 0.642i)9-s + (−0.326 − 0.118i)13-s + (−0.5 − 0.866i)19-s + (−1.43 − 1.20i)21-s + (−0.939 − 0.342i)25-s + (−0.500 + 0.866i)27-s + (−0.766 − 1.32i)31-s + 1.53·37-s + 0.347·39-s + (0.266 − 1.50i)43-s + (−1.26 + 2.19i)49-s + (0.766 + 0.642i)57-s + (0.0603 + 0.342i)61-s + ⋯
L(s)  = 1  + (−0.939 + 0.342i)3-s + (0.939 + 1.62i)7-s + (0.766 − 0.642i)9-s + (−0.326 − 0.118i)13-s + (−0.5 − 0.866i)19-s + (−1.43 − 1.20i)21-s + (−0.939 − 0.342i)25-s + (−0.500 + 0.866i)27-s + (−0.766 − 1.32i)31-s + 1.53·37-s + 0.347·39-s + (0.266 − 1.50i)43-s + (−1.26 + 2.19i)49-s + (0.766 + 0.642i)57-s + (0.0603 + 0.342i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 228 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.756 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 228 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.756 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(228\)    =    \(2^{2} \cdot 3 \cdot 19\)
Sign: $0.756 - 0.654i$
Analytic conductor: \(0.113786\)
Root analytic conductor: \(0.337323\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{228} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 228,\ (\ :0),\ 0.756 - 0.654i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5965716834\)
\(L(\frac12)\) \(\approx\) \(0.5965716834\)
\(L(1)\) 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 \( 1 + (0.939 - 0.342i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
good5 \( 1 + (0.939 + 0.342i)T^{2} \)
7 \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.326 + 0.118i)T + (0.766 + 0.642i)T^{2} \)
17 \( 1 + (-0.173 - 0.984i)T^{2} \)
23 \( 1 + (0.939 - 0.342i)T^{2} \)
29 \( 1 + (-0.173 + 0.984i)T^{2} \)
31 \( 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 - 1.53T + T^{2} \)
41 \( 1 + (-0.766 + 0.642i)T^{2} \)
43 \( 1 + (-0.266 + 1.50i)T + (-0.939 - 0.342i)T^{2} \)
47 \( 1 + (-0.173 + 0.984i)T^{2} \)
53 \( 1 + (0.939 - 0.342i)T^{2} \)
59 \( 1 + (-0.173 - 0.984i)T^{2} \)
61 \( 1 + (-0.0603 - 0.342i)T + (-0.939 + 0.342i)T^{2} \)
67 \( 1 + (1.43 - 1.20i)T + (0.173 - 0.984i)T^{2} \)
71 \( 1 + (0.939 + 0.342i)T^{2} \)
73 \( 1 + (0.326 - 0.118i)T + (0.766 - 0.642i)T^{2} \)
79 \( 1 + (-1.76 + 0.642i)T + (0.766 - 0.642i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T^{2} \)
89 \( 1 + (-0.766 - 0.642i)T^{2} \)
97 \( 1 + (0.766 + 0.642i)T + (0.173 + 0.984i)T^{2} \)
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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

−12.24178980894681560527179094947, −11.62908781533890540191818559849, −10.90268483140256547162961089141, −9.644868190922176379766015402785, −8.797824659548627564189731178324, −7.58635044232470429388292242994, −6.11849197904606841331446093956, −5.39321199046958206380717898193, −4.34081692186315113372638408770, −2.27874071348170025039357565269, 1.50982751497776689218005851643, 4.02634525145321782775232691616, 4.95923144033880540540075629516, 6.28481136834883854916484166649, 7.38857740287814404294217850660, 7.990945917030257619542676595666, 9.755476444356326887659613267974, 10.68841063034004938204044037024, 11.22705388470360905952556821228, 12.27241640337287264413991556277

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