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Properties

Label 2-228-12.11-c1-0-15
Degree $2$
Conductor $228$
Sign $-0.253 - 0.967i$
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  + (1.13 + 0.841i)2-s + (−0.0680 + 1.73i)3-s + (0.583 + 1.91i)4-s − 1.36i·5-s + (−1.53 + 1.90i)6-s + 1.12i·7-s + (−0.946 + 2.66i)8-s + (−2.99 − 0.235i)9-s + (1.15 − 1.55i)10-s + 3.43·11-s + (−3.35 + 0.879i)12-s − 0.802·13-s + (−0.950 + 1.28i)14-s + (2.36 + 0.0930i)15-s + (−3.31 + 2.23i)16-s − 5.04i·17-s + ⋯
L(s)  = 1  + (0.803 + 0.595i)2-s + (−0.0392 + 0.999i)3-s + (0.291 + 0.956i)4-s − 0.611i·5-s + (−0.626 + 0.779i)6-s + 0.426i·7-s + (−0.334 + 0.942i)8-s + (−0.996 − 0.0784i)9-s + (0.364 − 0.491i)10-s + 1.03·11-s + (−0.967 + 0.253i)12-s − 0.222·13-s + (−0.254 + 0.343i)14-s + (0.611 + 0.0240i)15-s + (−0.829 + 0.558i)16-s − 1.22i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 228 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.253 - 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 228 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.253 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(228\)    =    \(2^{2} \cdot 3 \cdot 19\)
Sign: $-0.253 - 0.967i$
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: \((2,\ 228,\ (\ :1/2),\ -0.253 - 0.967i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.10425 + 1.43160i\)
\(L(\frac12)\) \(\approx\) \(1.10425 + 1.43160i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.13 - 0.841i)T \)
3 \( 1 + (0.0680 - 1.73i)T \)
19 \( 1 + iT \)
good5 \( 1 + 1.36iT - 5T^{2} \)
7 \( 1 - 1.12iT - 7T^{2} \)
11 \( 1 - 3.43T + 11T^{2} \)
13 \( 1 + 0.802T + 13T^{2} \)
17 \( 1 + 5.04iT - 17T^{2} \)
23 \( 1 - 0.107T + 23T^{2} \)
29 \( 1 + 2.28iT - 29T^{2} \)
31 \( 1 - 6.32iT - 31T^{2} \)
37 \( 1 - 4.86T + 37T^{2} \)
41 \( 1 + 7.80iT - 41T^{2} \)
43 \( 1 + 4.40iT - 43T^{2} \)
47 \( 1 - 5.92T + 47T^{2} \)
53 \( 1 - 7.95iT - 53T^{2} \)
59 \( 1 + 14.9T + 59T^{2} \)
61 \( 1 + 11.9T + 61T^{2} \)
67 \( 1 + 8.03iT - 67T^{2} \)
71 \( 1 + 15.4T + 71T^{2} \)
73 \( 1 + 0.852T + 73T^{2} \)
79 \( 1 - 9.56iT - 79T^{2} \)
83 \( 1 + 6.54T + 83T^{2} \)
89 \( 1 + 4.58iT - 89T^{2} \)
97 \( 1 - 15.8T + 97T^{2} \)
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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

−12.31387036055149427736899673714, −11.85753434144117198352765905226, −10.73593208793525397415673874858, −9.176507560091534968600556009793, −8.853165369188141383889252622686, −7.37219374047007077325075833315, −6.08255760862981568391007082377, −5.06280097777269245482200804417, −4.25925600198491797122307987622, −2.90952918732031091818665239739, 1.47233502635739910647707305915, 2.96550401063208625687308615373, 4.26078286522817048036131722101, 5.94942476149116381272405862703, 6.59866936460487682319855732144, 7.66143796676294990466235346620, 9.124311437261044922981817957499, 10.39769036161019690250726247198, 11.21536599347825376451309533946, 12.02863618396870683230403015327

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