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Properties

Label 2-228-57.5-c0-0-0
Degree $2$
Conductor $228$
Sign $0.672 + 0.740i$
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  + (0.173 − 0.984i)3-s + (−0.173 − 0.300i)7-s + (−0.939 − 0.342i)9-s + (0.266 + 1.50i)13-s + (−0.5 − 0.866i)19-s + (−0.326 + 0.118i)21-s + (0.173 + 0.984i)25-s + (−0.5 + 0.866i)27-s + (0.939 + 1.62i)31-s − 1.87·37-s + 1.53·39-s + (−1.43 − 1.20i)43-s + (0.439 − 0.761i)49-s + (−0.939 + 0.342i)57-s + (1.17 − 0.984i)61-s + ⋯
L(s)  = 1  + (0.173 − 0.984i)3-s + (−0.173 − 0.300i)7-s + (−0.939 − 0.342i)9-s + (0.266 + 1.50i)13-s + (−0.5 − 0.866i)19-s + (−0.326 + 0.118i)21-s + (0.173 + 0.984i)25-s + (−0.5 + 0.866i)27-s + (0.939 + 1.62i)31-s − 1.87·37-s + 1.53·39-s + (−1.43 − 1.20i)43-s + (0.439 − 0.761i)49-s + (−0.939 + 0.342i)57-s + (1.17 − 0.984i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 228 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.672 + 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 228 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.672 + 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(228\)    =    \(2^{2} \cdot 3 \cdot 19\)
Sign: $0.672 + 0.740i$
Analytic conductor: \(0.113786\)
Root analytic conductor: \(0.337323\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{228} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 228,\ (\ :0),\ 0.672 + 0.740i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7311324414\)
\(L(\frac12)\) \(\approx\) \(0.7311324414\)
\(L(1)\) 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 \( 1 + (-0.173 + 0.984i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
good5 \( 1 + (-0.173 - 0.984i)T^{2} \)
7 \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.266 - 1.50i)T + (-0.939 + 0.342i)T^{2} \)
17 \( 1 + (-0.766 + 0.642i)T^{2} \)
23 \( 1 + (-0.173 + 0.984i)T^{2} \)
29 \( 1 + (-0.766 - 0.642i)T^{2} \)
31 \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + 1.87T + T^{2} \)
41 \( 1 + (0.939 + 0.342i)T^{2} \)
43 \( 1 + (1.43 + 1.20i)T + (0.173 + 0.984i)T^{2} \)
47 \( 1 + (-0.766 - 0.642i)T^{2} \)
53 \( 1 + (-0.173 + 0.984i)T^{2} \)
59 \( 1 + (-0.766 + 0.642i)T^{2} \)
61 \( 1 + (-1.17 + 0.984i)T + (0.173 - 0.984i)T^{2} \)
67 \( 1 + (0.326 + 0.118i)T + (0.766 + 0.642i)T^{2} \)
71 \( 1 + (-0.173 - 0.984i)T^{2} \)
73 \( 1 + (-0.266 + 1.50i)T + (-0.939 - 0.342i)T^{2} \)
79 \( 1 + (-0.0603 + 0.342i)T + (-0.939 - 0.342i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T^{2} \)
89 \( 1 + (0.939 - 0.342i)T^{2} \)
97 \( 1 + (-0.939 + 0.342i)T + (0.766 - 0.642i)T^{2} \)
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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

−12.28528218320688975514236439800, −11.58769298921265296854708318224, −10.53494170561631309742673702196, −9.126880305318087456337997934159, −8.468106292805154241575351725615, −7.00352180687488904356497950034, −6.65931453732311154453761455005, −5.06868025867360532327881455005, −3.46422594205337024592826052817, −1.82463447928634437616145705469, 2.76015719602775940537678612997, 3.97661258265047751708350429215, 5.28333195368743484907088279275, 6.22429988363595210209764897536, 7.973323581145115931312157885382, 8.624951480029773396153102792986, 9.931580349991353123162727530058, 10.39715299150644739886753440407, 11.51502863089367685611728516470, 12.54675094406521797936147283870

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