Properties

Label 2-572-11.3-c1-0-5
Degree $2$
Conductor $572$
Sign $0.955 + 0.293i$
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  + (−0.146 − 0.452i)3-s + (3.01 + 2.18i)5-s + (0.736 − 2.26i)7-s + (2.24 − 1.63i)9-s + (−2.05 − 2.60i)11-s + (−0.809 + 0.587i)13-s + (0.546 − 1.68i)15-s + (1.68 + 1.22i)17-s + (0.128 + 0.394i)19-s − 1.13·21-s + 2.03·23-s + (2.74 + 8.43i)25-s + (−2.22 − 1.61i)27-s + (−1.36 + 4.19i)29-s + (0.681 − 0.495i)31-s + ⋯
L(s)  = 1  + (−0.0848 − 0.260i)3-s + (1.34 + 0.978i)5-s + (0.278 − 0.856i)7-s + (0.748 − 0.543i)9-s + (−0.619 − 0.785i)11-s + (−0.224 + 0.163i)13-s + (0.141 − 0.434i)15-s + (0.408 + 0.296i)17-s + (0.0293 + 0.0904i)19-s − 0.247·21-s + 0.424·23-s + (0.548 + 1.68i)25-s + (−0.427 − 0.310i)27-s + (−0.253 + 0.779i)29-s + (0.122 − 0.0889i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.955 + 0.293i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.955 + 0.293i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(572\)    =    \(2^{2} \cdot 11 \cdot 13\)
Sign: $0.955 + 0.293i$
Analytic conductor: \(4.56744\)
Root analytic conductor: \(2.13715\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{572} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 572,\ (\ :1/2),\ 0.955 + 0.293i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.84473 - 0.277183i\)
\(L(\frac12)\) \(\approx\) \(1.84473 - 0.277183i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.05 + 2.60i)T \)
13 \( 1 + (0.809 - 0.587i)T \)
good3 \( 1 + (0.146 + 0.452i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (-3.01 - 2.18i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (-0.736 + 2.26i)T + (-5.66 - 4.11i)T^{2} \)
17 \( 1 + (-1.68 - 1.22i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-0.128 - 0.394i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 2.03T + 23T^{2} \)
29 \( 1 + (1.36 - 4.19i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-0.681 + 0.495i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-3.25 + 10.0i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-0.668 - 2.05i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 4.76T + 43T^{2} \)
47 \( 1 + (-3.01 - 9.27i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-4.01 + 2.91i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-1.06 + 3.27i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (11.0 + 8.04i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 1.00T + 67T^{2} \)
71 \( 1 + (-6.70 - 4.87i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (4.91 - 15.1i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (3.27 - 2.37i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (0.759 + 0.551i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 10.1T + 89T^{2} \)
97 \( 1 + (8.95 - 6.50i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.63825447405790788841754463600, −9.960399116221184299186827688995, −9.174394663386307414728861644256, −7.78065397217603516178725908312, −7.01612788395341185650439585222, −6.23239086315120479059957660284, −5.34394175445328328211752354364, −3.88301901311932427308921793098, −2.68315494760930446930480229886, −1.32656504755212536751542160794, 1.59537997952688112322621652151, 2.55916460402514012201252463957, 4.58019886466415995320995160615, 5.15797860594388101437656248546, 5.89870488310867246298722470283, 7.22837828490699461389185988215, 8.283871793295523487592888350470, 9.181193367324407707150500442350, 9.910471645123956920756556003540, 10.41109685992384822148415083801

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