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Properties

Label 2-228-228.227-c2-0-50
Degree $2$
Conductor $228$
Sign $0.713 + 0.700i$
Analytic cond. $6.21255$
Root an. cond. $2.49249$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.56 + 1.25i)2-s + (2.94 + 0.566i)3-s + (0.867 − 3.90i)4-s − 9.23i·5-s + (−5.30 + 2.80i)6-s + 4.75i·7-s + (3.53 + 7.17i)8-s + (8.35 + 3.33i)9-s + (11.5 + 14.4i)10-s − 10.8·11-s + (4.76 − 11.0i)12-s − 16.9i·13-s + (−5.95 − 7.42i)14-s + (5.22 − 27.2i)15-s + (−14.4 − 6.77i)16-s − 14.5i·17-s + ⋯
L(s)  = 1  + (−0.780 + 0.625i)2-s + (0.982 + 0.188i)3-s + (0.216 − 0.976i)4-s − 1.84i·5-s + (−0.884 + 0.467i)6-s + 0.679i·7-s + (0.441 + 0.897i)8-s + (0.928 + 0.370i)9-s + (1.15 + 1.44i)10-s − 0.986·11-s + (0.397 − 0.917i)12-s − 1.30i·13-s + (−0.425 − 0.530i)14-s + (0.348 − 1.81i)15-s + (−0.905 − 0.423i)16-s − 0.853i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 228 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.713 + 0.700i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 228 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.713 + 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(228\)    =    \(2^{2} \cdot 3 \cdot 19\)
Sign: $0.713 + 0.700i$
Analytic conductor: \(6.21255\)
Root analytic conductor: \(2.49249\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{228} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 228,\ (\ :1),\ 0.713 + 0.700i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.26885 - 0.519147i\)
\(L(\frac12)\) \(\approx\) \(1.26885 - 0.519147i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.56 - 1.25i)T \)
3 \( 1 + (-2.94 - 0.566i)T \)
19 \( 1 + (-6.84 + 17.7i)T \)
good5 \( 1 + 9.23iT - 25T^{2} \)
7 \( 1 - 4.75iT - 49T^{2} \)
11 \( 1 + 10.8T + 121T^{2} \)
13 \( 1 + 16.9iT - 169T^{2} \)
17 \( 1 + 14.5iT - 289T^{2} \)
23 \( 1 - 30.5T + 529T^{2} \)
29 \( 1 - 29.6T + 841T^{2} \)
31 \( 1 + 20.0T + 961T^{2} \)
37 \( 1 - 38.4iT - 1.36e3T^{2} \)
41 \( 1 + 55.0T + 1.68e3T^{2} \)
43 \( 1 + 22.6iT - 1.84e3T^{2} \)
47 \( 1 - 18.4T + 2.20e3T^{2} \)
53 \( 1 - 32.2T + 2.80e3T^{2} \)
59 \( 1 - 41.8iT - 3.48e3T^{2} \)
61 \( 1 - 44.2T + 3.72e3T^{2} \)
67 \( 1 + 23.5T + 4.48e3T^{2} \)
71 \( 1 - 90.1iT - 5.04e3T^{2} \)
73 \( 1 + 30.0T + 5.32e3T^{2} \)
79 \( 1 - 50.6T + 6.24e3T^{2} \)
83 \( 1 - 36.9T + 6.88e3T^{2} \)
89 \( 1 - 70.6T + 7.92e3T^{2} \)
97 \( 1 - 84.1iT - 9.40e3T^{2} \)
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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

−11.97782197089648864972589019312, −10.46668552445173722268279987769, −9.507156997546226472815718344933, −8.749672970336054884212723603350, −8.276660035039384657701338603751, −7.27293278158316745069639221928, −5.30633160872231024397657186277, −4.96206972374199910233940974309, −2.65581820794071799972078631889, −0.891763408603422272735201734017, 1.94363559142750041467725121333, 3.06632581985481036271342937328, 3.94261831556883197320147658909, 6.65705831580305062715656996868, 7.27233243372468170831413692852, 8.097960583449871569802723059414, 9.343981022983885846770863947173, 10.38473457858976592543905818209, 10.70387435415872414459519007449, 11.89831337854142921987814484264

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