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Properties

Label 2-228-12.11-c1-0-21
Degree $2$
Conductor $228$
Sign $0.865 - 0.500i$
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  + (1.41 − 0.0507i)2-s + (0.756 + 1.55i)3-s + (1.99 − 0.143i)4-s + 0.762i·5-s + (1.14 + 2.16i)6-s − 3.87i·7-s + (2.81 − 0.303i)8-s + (−1.85 + 2.35i)9-s + (0.0386 + 1.07i)10-s − 5.11·11-s + (1.73 + 2.99i)12-s − 0.587·13-s + (−0.196 − 5.47i)14-s + (−1.18 + 0.577i)15-s + (3.95 − 0.572i)16-s + 1.64i·17-s + ⋯
L(s)  = 1  + (0.999 − 0.0358i)2-s + (0.436 + 0.899i)3-s + (0.997 − 0.0716i)4-s + 0.341i·5-s + (0.468 + 0.883i)6-s − 1.46i·7-s + (0.994 − 0.107i)8-s + (−0.618 + 0.786i)9-s + (0.0122 + 0.340i)10-s − 1.54·11-s + (0.500 + 0.865i)12-s − 0.162·13-s + (−0.0525 − 1.46i)14-s + (−0.306 + 0.149i)15-s + (0.989 − 0.143i)16-s + 0.399i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 228 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.865 - 0.500i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 228 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.865 - 0.500i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(228\)    =    \(2^{2} \cdot 3 \cdot 19\)
Sign: $0.865 - 0.500i$
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: \((2,\ 228,\ (\ :1/2),\ 0.865 - 0.500i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.23081 + 0.598145i\)
\(L(\frac12)\) \(\approx\) \(2.23081 + 0.598145i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.41 + 0.0507i)T \)
3 \( 1 + (-0.756 - 1.55i)T \)
19 \( 1 - iT \)
good5 \( 1 - 0.762iT - 5T^{2} \)
7 \( 1 + 3.87iT - 7T^{2} \)
11 \( 1 + 5.11T + 11T^{2} \)
13 \( 1 + 0.587T + 13T^{2} \)
17 \( 1 - 1.64iT - 17T^{2} \)
23 \( 1 + 4.13T + 23T^{2} \)
29 \( 1 - 0.844iT - 29T^{2} \)
31 \( 1 + 8.95iT - 31T^{2} \)
37 \( 1 + 1.82T + 37T^{2} \)
41 \( 1 + 4.99iT - 41T^{2} \)
43 \( 1 - 8.79iT - 43T^{2} \)
47 \( 1 - 10.6T + 47T^{2} \)
53 \( 1 - 10.3iT - 53T^{2} \)
59 \( 1 + 7.90T + 59T^{2} \)
61 \( 1 - 3.61T + 61T^{2} \)
67 \( 1 + 8.43iT - 67T^{2} \)
71 \( 1 - 5.12T + 71T^{2} \)
73 \( 1 + 7.37T + 73T^{2} \)
79 \( 1 - 13.4iT - 79T^{2} \)
83 \( 1 - 6.07T + 83T^{2} \)
89 \( 1 - 9.06iT - 89T^{2} \)
97 \( 1 + 4.70T + 97T^{2} \)
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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

−12.52104422977892819749721346048, −10.97665989885400598930750805072, −10.63407557024522984428709994480, −9.840886916270173175611470741707, −8.019419662292114713099328532332, −7.35580704939213299867410551518, −5.86170117689759635349646949084, −4.66332078147602917062733229251, −3.79475278999384044807722790740, −2.58293358051079878853889214830, 2.17572411468790641293580597094, 3.04317355608096736650254996871, 5.03553990202619568407168316378, 5.78068422987774288259678035255, 6.99294357228736082390534794176, 8.062021462368124249071549013699, 8.897292053588112032431916230844, 10.42636151535657914800202112000, 11.73141112596829784054339512516, 12.36422194416323201589575680063

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