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Properties

Label 2-228-12.11-c1-0-3
Degree $2$
Conductor $228$
Sign $-0.837 - 0.546i$
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  + (0.791 + 1.17i)2-s + (−1.69 − 0.337i)3-s + (−0.747 + 1.85i)4-s − 0.951i·5-s + (−0.949 − 2.25i)6-s + 4.82i·7-s + (−2.76 + 0.592i)8-s + (2.77 + 1.14i)9-s + (1.11 − 0.752i)10-s − 3.58·11-s + (1.89 − 2.90i)12-s − 2.89·13-s + (−5.65 + 3.81i)14-s + (−0.320 + 1.61i)15-s + (−2.88 − 2.77i)16-s + 3.29i·17-s + ⋯
L(s)  = 1  + (0.559 + 0.828i)2-s + (−0.980 − 0.194i)3-s + (−0.373 + 0.927i)4-s − 0.425i·5-s + (−0.387 − 0.921i)6-s + 1.82i·7-s + (−0.977 + 0.209i)8-s + (0.924 + 0.381i)9-s + (0.352 − 0.238i)10-s − 1.08·11-s + (0.546 − 0.837i)12-s − 0.801·13-s + (−1.51 + 1.02i)14-s + (−0.0827 + 0.417i)15-s + (−0.720 − 0.692i)16-s + 0.799i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 228 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.837 - 0.546i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 228 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.837 - 0.546i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(228\)    =    \(2^{2} \cdot 3 \cdot 19\)
Sign: $-0.837 - 0.546i$
Analytic conductor: \(1.82058\)
Root analytic conductor: \(1.34929\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{228} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 228,\ (\ :1/2),\ -0.837 - 0.546i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.262259 + 0.880955i\)
\(L(\frac12)\) \(\approx\) \(0.262259 + 0.880955i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.791 - 1.17i)T \)
3 \( 1 + (1.69 + 0.337i)T \)
19 \( 1 - iT \)
good5 \( 1 + 0.951iT - 5T^{2} \)
7 \( 1 - 4.82iT - 7T^{2} \)
11 \( 1 + 3.58T + 11T^{2} \)
13 \( 1 + 2.89T + 13T^{2} \)
17 \( 1 - 3.29iT - 17T^{2} \)
23 \( 1 - 8.68T + 23T^{2} \)
29 \( 1 - 5.48iT - 29T^{2} \)
31 \( 1 + 6.59iT - 31T^{2} \)
37 \( 1 - 8.68T + 37T^{2} \)
41 \( 1 + 0.964iT - 41T^{2} \)
43 \( 1 - 3.20iT - 43T^{2} \)
47 \( 1 - 0.581T + 47T^{2} \)
53 \( 1 - 1.07iT - 53T^{2} \)
59 \( 1 - 4.73T + 59T^{2} \)
61 \( 1 + 5.55T + 61T^{2} \)
67 \( 1 - 2.99iT - 67T^{2} \)
71 \( 1 + 4.79T + 71T^{2} \)
73 \( 1 + 1.18T + 73T^{2} \)
79 \( 1 + 0.0661iT - 79T^{2} \)
83 \( 1 - 8.75T + 83T^{2} \)
89 \( 1 - 6.42iT - 89T^{2} \)
97 \( 1 + 0.00303T + 97T^{2} \)
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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

−12.79676750883697070423097283501, −12.00416854115929290667917854213, −10.99886774569717840921636058596, −9.481813164674569979584208538156, −8.510491977250157425789681232605, −7.46276642734913157982540346313, −6.21847034770672585675162381350, −5.37874076591566013178895757802, −4.79578665295991419466563571371, −2.66368742115383895128320044235, 0.72261066807118439909218154359, 3.02071585254564424503783526721, 4.47360844841642955035308285641, 5.13272147726088859510255632192, 6.69288457645035560839123593372, 7.42529434213634311566166675252, 9.496139218890900863469671881348, 10.41934568410170144589374200908, 10.78031639594122040046260477608, 11.62772261704846249455579772749

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