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Properties

Label 2-228-12.11-c1-0-6
Degree $2$
Conductor $228$
Sign $-0.0451 - 0.998i$
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  + (−1.17 + 0.788i)2-s + (0.726 + 1.57i)3-s + (0.755 − 1.85i)4-s − 2.76i·5-s + (−2.09 − 1.27i)6-s + 4.18i·7-s + (0.573 + 2.76i)8-s + (−1.94 + 2.28i)9-s + (2.18 + 3.24i)10-s + 1.17·11-s + (3.46 − 0.156i)12-s + 5.33·13-s + (−3.29 − 4.91i)14-s + (4.34 − 2.00i)15-s + (−2.85 − 2.79i)16-s + 6.95i·17-s + ⋯
L(s)  = 1  + (−0.829 + 0.557i)2-s + (0.419 + 0.907i)3-s + (0.377 − 0.925i)4-s − 1.23i·5-s + (−0.854 − 0.519i)6-s + 1.58i·7-s + (0.202 + 0.979i)8-s + (−0.648 + 0.761i)9-s + (0.689 + 1.02i)10-s + 0.354·11-s + (0.998 − 0.0451i)12-s + 1.48·13-s + (−0.881 − 1.31i)14-s + (1.12 − 0.518i)15-s + (−0.714 − 0.699i)16-s + 1.68i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 228 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0451 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 228 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0451 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(228\)    =    \(2^{2} \cdot 3 \cdot 19\)
Sign: $-0.0451 - 0.998i$
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: \((2,\ 228,\ (\ :1/2),\ -0.0451 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.666317 + 0.697113i\)
\(L(\frac12)\) \(\approx\) \(0.666317 + 0.697113i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.17 - 0.788i)T \)
3 \( 1 + (-0.726 - 1.57i)T \)
19 \( 1 - iT \)
good5 \( 1 + 2.76iT - 5T^{2} \)
7 \( 1 - 4.18iT - 7T^{2} \)
11 \( 1 - 1.17T + 11T^{2} \)
13 \( 1 - 5.33T + 13T^{2} \)
17 \( 1 - 6.95iT - 17T^{2} \)
23 \( 1 + 2.96T + 23T^{2} \)
29 \( 1 + 3.64iT - 29T^{2} \)
31 \( 1 + 4.24iT - 31T^{2} \)
37 \( 1 + 2.32T + 37T^{2} \)
41 \( 1 + 3.23iT - 41T^{2} \)
43 \( 1 + 8.22iT - 43T^{2} \)
47 \( 1 - 7.26T + 47T^{2} \)
53 \( 1 + 0.986iT - 53T^{2} \)
59 \( 1 + 2.71T + 59T^{2} \)
61 \( 1 - 11.4T + 61T^{2} \)
67 \( 1 + 0.752iT - 67T^{2} \)
71 \( 1 - 4.98T + 71T^{2} \)
73 \( 1 + 5.51T + 73T^{2} \)
79 \( 1 + 3.62iT - 79T^{2} \)
83 \( 1 + 8.74T + 83T^{2} \)
89 \( 1 + 7.44iT - 89T^{2} \)
97 \( 1 + 3.52T + 97T^{2} \)
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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

−12.32855321545929750285019867904, −11.31749261721747052210887788032, −10.22676458153700991587263687315, −9.166900879050824154734157289825, −8.604076370969530979217057374361, −8.220388165287790444884832839817, −6.01437971158290310128876562201, −5.52158244094175869412497178512, −4.04347283450148903096736641890, −1.93081168714950833262690462998, 1.12639709766184327349333688954, 2.89614197185674649382662732200, 3.80697352964691177253281202785, 6.57224021382314576000167880180, 7.04831137906654030323677961638, 7.87816708377934084245278550823, 9.043887850536630529335212922996, 10.17962642174622784382309433977, 11.03924652827171903447104394508, 11.60988131192125794470193618785

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