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Properties

Label 2-572-13.12-c1-0-0
Degree $2$
Conductor $572$
Sign $-0.998 + 0.0617i$
Analytic cond. $4.56744$
Root an. cond. $2.13715$
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  − 1.43·3-s + 4.37i·5-s + 2.43i·7-s − 0.941·9-s + i·11-s + (−0.222 − 3.59i)13-s − 6.27i·15-s − 5.19·17-s − 4.25i·19-s − 3.49i·21-s + 6.81·23-s − 14.1·25-s + 5.65·27-s − 4.42·29-s − 1.62i·31-s + ⋯
L(s)  = 1  − 0.828·3-s + 1.95i·5-s + 0.920i·7-s − 0.313·9-s + 0.301i·11-s + (−0.0617 − 0.998i)13-s − 1.62i·15-s − 1.26·17-s − 0.976i·19-s − 0.762i·21-s + 1.42·23-s − 2.82·25-s + 1.08·27-s − 0.821·29-s − 0.291i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0617i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0617i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(572\)    =    \(2^{2} \cdot 11 \cdot 13\)
Sign: $-0.998 + 0.0617i$
Analytic conductor: \(4.56744\)
Root analytic conductor: \(2.13715\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{572} (441, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 572,\ (\ :1/2),\ -0.998 + 0.0617i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0160361 - 0.518713i\)
\(L(\frac12)\) \(\approx\) \(0.0160361 - 0.518713i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 - iT \)
13 \( 1 + (0.222 + 3.59i)T \)
good3 \( 1 + 1.43T + 3T^{2} \)
5 \( 1 - 4.37iT - 5T^{2} \)
7 \( 1 - 2.43iT - 7T^{2} \)
17 \( 1 + 5.19T + 17T^{2} \)
19 \( 1 + 4.25iT - 19T^{2} \)
23 \( 1 - 6.81T + 23T^{2} \)
29 \( 1 + 4.42T + 29T^{2} \)
31 \( 1 + 1.62iT - 31T^{2} \)
37 \( 1 - 3.50iT - 37T^{2} \)
41 \( 1 - 9.53iT - 41T^{2} \)
43 \( 1 + 6.42T + 43T^{2} \)
47 \( 1 - 1.13iT - 47T^{2} \)
53 \( 1 + 1.71T + 53T^{2} \)
59 \( 1 + 3.60iT - 59T^{2} \)
61 \( 1 + 8.42T + 61T^{2} \)
67 \( 1 - 8.01iT - 67T^{2} \)
71 \( 1 - 0.279iT - 71T^{2} \)
73 \( 1 - 6.07iT - 73T^{2} \)
79 \( 1 + 3.10T + 79T^{2} \)
83 \( 1 - 2.88iT - 83T^{2} \)
89 \( 1 - 6.11iT - 89T^{2} \)
97 \( 1 + 12.3iT - 97T^{2} \)
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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

−11.28238582666374930293663114358, −10.59151245019659369290794335693, −9.633154176328324900267862407711, −8.539071498035841095814474732627, −7.30941928228410600201337305378, −6.58227440393340183515491621515, −5.89019039468489486853115157163, −4.86877847792134952948895466476, −3.13786318747423411064850628201, −2.49037522587315393323109036326, 0.32031759698496543665892156220, 1.62390971772423844438212996444, 3.90788565597923771435762348815, 4.73857705102577970216810986047, 5.47420906972775338343142209186, 6.51558146041847777841481644763, 7.63358872518732178817403005631, 8.838510540039815429425886841138, 9.093392722493657322273631346187, 10.42755941376220991599395521315

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