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Properties

Label 2-228-19.8-c2-0-3
Degree $2$
Conductor $228$
Sign $0.995 - 0.0977i$
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  + (1.5 + 0.866i)3-s + (1 − 1.73i)5-s − 7-s + (1.5 + 2.59i)9-s + 16·11-s + (13.5 − 7.79i)13-s + (3 − 1.73i)15-s + (−11 + 19.0i)17-s + 19·19-s + (−1.5 − 0.866i)21-s + (−20 − 34.6i)23-s + (10.5 + 18.1i)25-s + 5.19i·27-s + (15 − 8.66i)29-s + 50.2i·31-s + ⋯
L(s)  = 1  + (0.5 + 0.288i)3-s + (0.200 − 0.346i)5-s − 0.142·7-s + (0.166 + 0.288i)9-s + 1.45·11-s + (1.03 − 0.599i)13-s + (0.200 − 0.115i)15-s + (−0.647 + 1.12i)17-s + 19-s + (−0.0714 − 0.0412i)21-s + (−0.869 − 1.50i)23-s + (0.419 + 0.727i)25-s + 0.192i·27-s + (0.517 − 0.298i)29-s + 1.62i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 228 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0977i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 228 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.995 - 0.0977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(228\)    =    \(2^{2} \cdot 3 \cdot 19\)
Sign: $0.995 - 0.0977i$
Analytic conductor: \(6.21255\)
Root analytic conductor: \(2.49249\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{228} (217, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 228,\ (\ :1),\ 0.995 - 0.0977i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.02653 + 0.0992702i\)
\(L(\frac12)\) \(\approx\) \(2.02653 + 0.0992702i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-1.5 - 0.866i)T \)
19 \( 1 - 19T \)
good5 \( 1 + (-1 + 1.73i)T + (-12.5 - 21.6i)T^{2} \)
7 \( 1 + T + 49T^{2} \)
11 \( 1 - 16T + 121T^{2} \)
13 \( 1 + (-13.5 + 7.79i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + (11 - 19.0i)T + (-144.5 - 250. i)T^{2} \)
23 \( 1 + (20 + 34.6i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (-15 + 8.66i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 - 50.2iT - 961T^{2} \)
37 \( 1 + 15.5iT - 1.36e3T^{2} \)
41 \( 1 + (12 + 6.92i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-24.5 + 42.4i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (23 + 39.8i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (42 - 24.2i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (57 + 32.9i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-48.5 - 84.0i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (22.5 - 12.9i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + (42 + 24.2i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (17.5 - 30.3i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (76.5 + 44.1i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 146T + 6.88e3T^{2} \)
89 \( 1 + (33 - 19.0i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (-54 - 31.1i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−12.10931683149161927560531041171, −10.92457264940197545839318516757, −10.04043919725702964297232537386, −8.868622697071046235476206118518, −8.446064498246633033403258436856, −6.88657719691798601487928393466, −5.87244992220180408910093251030, −4.38081014211925917088196726164, −3.33680341240190845421594917441, −1.46304830301414513428026097913, 1.47063226562894192984718442758, 3.12420560569280445443122636641, 4.31605659965606480652765252899, 6.09117126068293478539030208026, 6.85021764243956778025087138377, 8.002925463593753517517403096494, 9.243310308260031291279739273810, 9.680938032547247718881869956298, 11.34797008395468596471422866420, 11.73340007688584468668695380825

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