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Properties

Label 2-228-76.31-c1-0-19
Degree $2$
Conductor $228$
Sign $-0.984 + 0.173i$
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  + (0.501 − 1.32i)2-s + (−0.5 − 0.866i)3-s + (−1.49 − 1.32i)4-s + (−0.720 − 1.24i)5-s + (−1.39 + 0.227i)6-s − 1.30i·7-s + (−2.50 + 1.31i)8-s + (−0.499 + 0.866i)9-s + (−2.01 + 0.327i)10-s + 1.29i·11-s + (−0.399 + 1.95i)12-s + (−0.106 − 0.0617i)13-s + (−1.71 − 0.651i)14-s + (−0.720 + 1.24i)15-s + (0.483 + 3.97i)16-s + (−2.80 − 4.84i)17-s + ⋯
L(s)  = 1  + (0.354 − 0.935i)2-s + (−0.288 − 0.499i)3-s + (−0.748 − 0.662i)4-s + (−0.322 − 0.558i)5-s + (−0.569 + 0.0926i)6-s − 0.491i·7-s + (−0.885 + 0.465i)8-s + (−0.166 + 0.288i)9-s + (−0.636 + 0.103i)10-s + 0.389i·11-s + (−0.115 + 0.565i)12-s + (−0.0296 − 0.0171i)13-s + (−0.459 − 0.174i)14-s + (−0.186 + 0.322i)15-s + (0.120 + 0.992i)16-s + (−0.679 − 1.17i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 228 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 + 0.173i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 228 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.984 + 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(228\)    =    \(2^{2} \cdot 3 \cdot 19\)
Sign: $-0.984 + 0.173i$
Analytic conductor: \(1.82058\)
Root analytic conductor: \(1.34929\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{228} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 228,\ (\ :1/2),\ -0.984 + 0.173i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0890591 - 1.01721i\)
\(L(\frac12)\) \(\approx\) \(0.0890591 - 1.01721i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.501 + 1.32i)T \)
3 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (-3.87 + 1.98i)T \)
good5 \( 1 + (0.720 + 1.24i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + 1.30iT - 7T^{2} \)
11 \( 1 - 1.29iT - 11T^{2} \)
13 \( 1 + (0.106 + 0.0617i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.80 + 4.84i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (1.35 + 0.784i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.13 + 1.23i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 9.02T + 31T^{2} \)
37 \( 1 - 3.59iT - 37T^{2} \)
41 \( 1 + (-8.02 + 4.63i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.41 - 1.39i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.98 - 1.14i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (7.36 + 4.25i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.117 + 0.204i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.15 - 7.19i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.26 + 5.65i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.10 + 5.37i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-6.36 - 11.0i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.30 + 3.99i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 12.5iT - 83T^{2} \)
89 \( 1 + (-8.06 - 4.65i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.61 - 3.81i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.86102524173318382735442632491, −11.08415582421324374861810100441, −9.977245940021005593957473877633, −9.034544621883951417077668133065, −7.82847301261457895530778645532, −6.58215377686509169147323802568, −5.13698188254102535610969566656, −4.29326944861234488574649287922, −2.62574342229929018343632162174, −0.822335052398757428501855899287, 3.17698703845785096685460692093, 4.31759785422604429283222705554, 5.61060649063412614546476132842, 6.41030390382841772473603618803, 7.62608858339421900601743389691, 8.614025391698572747757964638057, 9.610092536112559620935468637014, 10.81548639776979062334007307351, 11.79360507748949267440176127811, 12.72122175914455288933913869934

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