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Properties

Label 2-572-572.571-c1-0-9
Degree $2$
Conductor $572$
Sign $0.679 - 0.733i$
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  + (0.516 − 1.31i)2-s + 1.51i·3-s + (−1.46 − 1.35i)4-s + (1.99 + 0.783i)6-s + 2.42i·7-s + (−2.54 + 1.22i)8-s + 0.697·9-s + 3.31i·11-s + (2.06 − 2.22i)12-s − 3.60·13-s + (3.19 + 1.25i)14-s + (0.304 + 3.98i)16-s + (0.359 − 0.917i)18-s + 5.06i·19-s − 3.67·21-s + (4.36 + 1.71i)22-s + ⋯
L(s)  = 1  + (0.365 − 0.930i)2-s + 0.876i·3-s + (−0.733 − 0.679i)4-s + (0.815 + 0.319i)6-s + 0.916i·7-s + (−0.900 + 0.434i)8-s + 0.232·9-s + 1.00i·11-s + (0.595 − 0.642i)12-s − 1.00·13-s + (0.852 + 0.334i)14-s + (0.0760 + 0.997i)16-s + (0.0848 − 0.216i)18-s + 1.16i·19-s − 0.802·21-s + (0.930 + 0.365i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.679 - 0.733i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.679 - 0.733i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(572\)    =    \(2^{2} \cdot 11 \cdot 13\)
Sign: $0.679 - 0.733i$
Analytic conductor: \(4.56744\)
Root analytic conductor: \(2.13715\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{572} (571, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 572,\ (\ :1/2),\ 0.679 - 0.733i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.28463 + 0.560995i\)
\(L(\frac12)\) \(\approx\) \(1.28463 + 0.560995i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.516 + 1.31i)T \)
11 \( 1 - 3.31iT \)
13 \( 1 + 3.60T \)
good3 \( 1 - 1.51iT - 3T^{2} \)
5 \( 1 - 5T^{2} \)
7 \( 1 - 2.42iT - 7T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 - 5.06iT - 19T^{2} \)
23 \( 1 + 0.711iT - 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 - 3.92T + 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 1.89T + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 13.2T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 17.1iT - 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−10.68605110358536923970239773839, −10.02076721545394418528200717129, −9.467729643806895740019141423626, −8.629757033171636992429755820713, −7.27914351389598136057096579081, −5.87378484691079076413500073765, −4.92461938716765473312664091654, −4.28525020139005817940838754546, −3.03340631038792807742471039076, −1.87590730601787830764774378366, 0.73191051963830214474175738247, 2.82221823925169393017967350331, 4.15415991923894338474079438782, 5.12007960690469227826929635296, 6.32930007719833328865074142836, 7.08056925742425032526167598729, 7.58968164772699413046838421409, 8.562905902476570465898181840063, 9.530513360977585774158980452018, 10.63835787129576578835159557975

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