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Properties

Label 2-572-11.3-c1-0-7
Degree $2$
Conductor $572$
Sign $0.954 - 0.299i$
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  + (0.525 + 1.61i)3-s + (1.11 + 0.808i)5-s + (1.47 − 4.53i)7-s + (0.0900 − 0.0653i)9-s + (3.16 + 0.981i)11-s + (0.809 − 0.587i)13-s + (−0.722 + 2.22i)15-s + (−1.47 − 1.07i)17-s + (−1.18 − 3.63i)19-s + 8.11·21-s − 0.328·23-s + (−0.960 − 2.95i)25-s + (4.27 + 3.10i)27-s + (−0.433 + 1.33i)29-s + (−5.49 + 3.99i)31-s + ⋯
L(s)  = 1  + (0.303 + 0.933i)3-s + (0.497 + 0.361i)5-s + (0.557 − 1.71i)7-s + (0.0300 − 0.0217i)9-s + (0.955 + 0.295i)11-s + (0.224 − 0.163i)13-s + (−0.186 + 0.574i)15-s + (−0.358 − 0.260i)17-s + (−0.271 − 0.834i)19-s + 1.76·21-s − 0.0685·23-s + (−0.192 − 0.591i)25-s + (0.823 + 0.598i)27-s + (−0.0804 + 0.247i)29-s + (−0.986 + 0.716i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.954 - 0.299i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.954 - 0.299i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(572\)    =    \(2^{2} \cdot 11 \cdot 13\)
Sign: $0.954 - 0.299i$
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: \((2,\ 572,\ (\ :1/2),\ 0.954 - 0.299i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.93272 + 0.296193i\)
\(L(\frac12)\) \(\approx\) \(1.93272 + 0.296193i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.16 - 0.981i)T \)
13 \( 1 + (-0.809 + 0.587i)T \)
good3 \( 1 + (-0.525 - 1.61i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (-1.11 - 0.808i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (-1.47 + 4.53i)T + (-5.66 - 4.11i)T^{2} \)
17 \( 1 + (1.47 + 1.07i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.18 + 3.63i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 0.328T + 23T^{2} \)
29 \( 1 + (0.433 - 1.33i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (5.49 - 3.99i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (2.47 - 7.62i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-3.66 - 11.2i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 3.92T + 43T^{2} \)
47 \( 1 + (-1.98 - 6.11i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (10.5 - 7.66i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-2.60 + 8.00i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (8.01 + 5.82i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 15.9T + 67T^{2} \)
71 \( 1 + (4.65 + 3.38i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (0.392 - 1.20i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (0.526 - 0.382i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (5.04 + 3.66i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 5.75T + 89T^{2} \)
97 \( 1 + (9.05 - 6.57i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.81971165010706750874614382463, −9.840395885196089630217835075960, −9.350704942025684664661737663323, −8.178606453454316790253280830651, −7.07449608211740474962127261604, −6.44477941528790190625841665345, −4.75116462518583765802332267297, −4.23967719033536842865967321594, −3.19099020613177504291655224162, −1.36611841340927584600075933266, 1.65511179492612120467739442009, 2.25501063434269152914965504015, 3.96538071404576276328755651853, 5.47360017497327434112250119261, 5.99615080207602761470539587794, 7.12884543469651451325150697519, 8.185631474675507966017069099575, 8.871663263593090411961197145997, 9.448703992800418816876391751581, 10.86115243519868113485823076465

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