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Properties

Label 2-572-572.571-c1-0-76
Degree $2$
Conductor $572$
Sign $-0.487 - 0.873i$
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  + (−0.356 − 1.36i)2-s − 3.05i·3-s + (−1.74 + 0.975i)4-s + (−4.18 + 1.08i)6-s − 5.22i·7-s + (1.95 + 2.04i)8-s − 6.35·9-s − 3.31i·11-s + (2.98 + 5.33i)12-s + 3.60·13-s + (−7.14 + 1.86i)14-s + (2.09 − 3.40i)16-s + (2.26 + 8.69i)18-s − 0.0771i·19-s − 15.9·21-s + (−4.53 + 1.18i)22-s + ⋯
L(s)  = 1  + (−0.251 − 0.967i)2-s − 1.76i·3-s + (−0.873 + 0.487i)4-s + (−1.70 + 0.444i)6-s − 1.97i·7-s + (0.691 + 0.722i)8-s − 2.11·9-s − 1.00i·11-s + (0.860 + 1.54i)12-s + 1.00·13-s + (−1.91 + 0.497i)14-s + (0.524 − 0.851i)16-s + (0.533 + 2.04i)18-s − 0.0177i·19-s − 3.48·21-s + (−0.967 + 0.251i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.487 - 0.873i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.487 - 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(572\)    =    \(2^{2} \cdot 11 \cdot 13\)
Sign: $-0.487 - 0.873i$
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: \((2,\ 572,\ (\ :1/2),\ -0.487 - 0.873i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.547569 + 0.932960i\)
\(L(\frac12)\) \(\approx\) \(0.547569 + 0.932960i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.356 + 1.36i)T \)
11 \( 1 + 3.31iT \)
13 \( 1 - 3.60T \)
good3 \( 1 + 3.05iT - 3T^{2} \)
5 \( 1 - 5T^{2} \)
7 \( 1 + 5.22iT - 7T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 + 0.0771iT - 19T^{2} \)
23 \( 1 - 8.87iT - 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 - 10.3T + 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 10.0T + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 4.44T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 0.671iT - 83T^{2} \)
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

−10.52699458327273428552878076969, −9.201782045430306534222337367381, −8.202970281968457388433860726105, −7.58731809330431771818369536543, −6.81977317708517863793544427942, −5.64508328338551183340769625731, −3.98314198415891752599339881122, −3.07441630515198520899849631887, −1.41458964978853597042199183405, −0.76597355273803042660232971525, 2.65097690762367857089251615230, 4.13213176400057683835853749568, 4.94673572254372621021782981644, 5.69494625435469042667249417298, 6.52236910851880442294572045279, 8.207400395175385890006847809360, 8.895215815904729222914936028422, 9.258407558709722155018406244054, 10.21673273087840658271881926599, 10.94382555295300678786096805669

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