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Properties

Label 2-228-12.11-c1-0-5
Degree $2$
Conductor $228$
Sign $0.928 - 0.371i$
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  + (−1.41 − 0.0507i)2-s + (−0.756 − 1.55i)3-s + (1.99 + 0.143i)4-s + 0.762i·5-s + (0.990 + 2.24i)6-s + 3.87i·7-s + (−2.81 − 0.303i)8-s + (−1.85 + 2.35i)9-s + (0.0386 − 1.07i)10-s + 5.11·11-s + (−1.28 − 3.21i)12-s − 0.587·13-s + (0.196 − 5.47i)14-s + (1.18 − 0.577i)15-s + (3.95 + 0.572i)16-s + 1.64i·17-s + ⋯
L(s)  = 1  + (−0.999 − 0.0358i)2-s + (−0.436 − 0.899i)3-s + (0.997 + 0.0716i)4-s + 0.341i·5-s + (0.404 + 0.914i)6-s + 1.46i·7-s + (−0.994 − 0.107i)8-s + (−0.618 + 0.786i)9-s + (0.0122 − 0.340i)10-s + 1.54·11-s + (−0.371 − 0.928i)12-s − 0.162·13-s + (0.0525 − 1.46i)14-s + (0.306 − 0.149i)15-s + (0.989 + 0.143i)16-s + 0.399i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 228 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.928 - 0.371i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 228 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.928 - 0.371i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(228\)    =    \(2^{2} \cdot 3 \cdot 19\)
Sign: $0.928 - 0.371i$
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: \((2,\ 228,\ (\ :1/2),\ 0.928 - 0.371i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.695658 + 0.133948i\)
\(L(\frac12)\) \(\approx\) \(0.695658 + 0.133948i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 + 0.0507i)T \)
3 \( 1 + (0.756 + 1.55i)T \)
19 \( 1 + iT \)
good5 \( 1 - 0.762iT - 5T^{2} \)
7 \( 1 - 3.87iT - 7T^{2} \)
11 \( 1 - 5.11T + 11T^{2} \)
13 \( 1 + 0.587T + 13T^{2} \)
17 \( 1 - 1.64iT - 17T^{2} \)
23 \( 1 - 4.13T + 23T^{2} \)
29 \( 1 - 0.844iT - 29T^{2} \)
31 \( 1 - 8.95iT - 31T^{2} \)
37 \( 1 + 1.82T + 37T^{2} \)
41 \( 1 + 4.99iT - 41T^{2} \)
43 \( 1 + 8.79iT - 43T^{2} \)
47 \( 1 + 10.6T + 47T^{2} \)
53 \( 1 - 10.3iT - 53T^{2} \)
59 \( 1 - 7.90T + 59T^{2} \)
61 \( 1 - 3.61T + 61T^{2} \)
67 \( 1 - 8.43iT - 67T^{2} \)
71 \( 1 + 5.12T + 71T^{2} \)
73 \( 1 + 7.37T + 73T^{2} \)
79 \( 1 + 13.4iT - 79T^{2} \)
83 \( 1 + 6.07T + 83T^{2} \)
89 \( 1 - 9.06iT - 89T^{2} \)
97 \( 1 + 4.70T + 97T^{2} \)
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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

−12.05079605336971899492398113719, −11.45605349694543849266848016715, −10.44144135141638283090286654210, −8.952935188589468165497345699268, −8.654531650384730498319921313749, −7.11653305585202619394497672964, −6.51573485656266812871331498037, −5.44509454888881696852619661174, −2.93696412479749077873604078922, −1.58350274455426834847423609305, 0.956304547893426331423152976029, 3.49892212581932180256245442121, 4.67018217521987830826940017448, 6.28769776472160158956106081966, 7.12541919759867182205478940604, 8.446005892986830966287365585156, 9.512795840734665640769201109513, 9.996824154636773495187990125164, 11.18759049970441074315695887314, 11.55863968292252320699142761101

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