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Properties

Label 2-228-57.8-c1-0-4
Degree $2$
Conductor $228$
Sign $-0.952 - 0.305i$
Analytic cond. $1.82058$
Root an. cond. $1.34929$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.5 − 0.866i)3-s − 5·7-s + (1.5 + 2.59i)9-s + (−4.5 + 2.59i)13-s + (−4 − 1.73i)19-s + (7.5 + 4.33i)21-s + (−2.5 − 4.33i)25-s − 5.19i·27-s + 8.66i·31-s − 5.19i·37-s + 9·39-s + (6.5 − 11.2i)43-s + 18·49-s + (4.5 + 6.06i)57-s + (−0.5 − 0.866i)61-s + ⋯
L(s)  = 1  + (−0.866 − 0.499i)3-s − 1.88·7-s + (0.5 + 0.866i)9-s + (−1.24 + 0.720i)13-s + (−0.917 − 0.397i)19-s + (1.63 + 0.944i)21-s + (−0.5 − 0.866i)25-s − 0.999i·27-s + 1.55i·31-s − 0.854i·37-s + 1.44·39-s + (0.991 − 1.71i)43-s + 2.57·49-s + (0.596 + 0.802i)57-s + (−0.0640 − 0.110i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 228 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.952 - 0.305i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 228 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.952 - 0.305i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(228\)    =    \(2^{2} \cdot 3 \cdot 19\)
Sign: $-0.952 - 0.305i$
Analytic conductor: \(1.82058\)
Root analytic conductor: \(1.34929\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{228} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 228,\ (\ :1/2),\ -0.952 - 0.305i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1.5 + 0.866i)T \)
19 \( 1 + (4 + 1.73i)T \)
good5 \( 1 + (2.5 + 4.33i)T^{2} \)
7 \( 1 + 5T + 7T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + (4.5 - 2.59i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 8.66iT - 31T^{2} \)
37 \( 1 + 5.19iT - 37T^{2} \)
41 \( 1 + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-6.5 + 11.2i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (10.5 - 6.06i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (8.5 - 14.7i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.5 + 2.59i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (12 + 6.92i)T + (48.5 + 84.0i)T^{2} \)
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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

−12.04201154941349638477142678930, −10.63449001845275867671211080365, −9.931643418350333875369967605454, −8.887804814931416945255008589054, −7.20942662953002707868174072265, −6.67930668148245683984310915948, −5.65532936585661186980171838388, −4.22478096844127391068925617324, −2.49267706559491071174247924841, 0, 3.00677681236077986908051424497, 4.28360662784915860997755102290, 5.71299791251531341421742039231, 6.44191281389796913815669196940, 7.57583103918023802549706842705, 9.342262533863547274433536128770, 9.842365737202050300264926135144, 10.66141606790894675708216854284, 11.89297722401288184521283510100

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