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Properties

Label 2-572-11.3-c1-0-10
Degree $2$
Conductor $572$
Sign $-0.525 + 0.850i$
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  + (0.955 + 2.93i)3-s + (−3.52 − 2.55i)5-s + (0.357 − 1.10i)7-s + (−5.30 + 3.85i)9-s + (−3.03 + 1.33i)11-s + (0.809 − 0.587i)13-s + (4.15 − 12.8i)15-s + (−6.11 − 4.44i)17-s + (−1.75 − 5.38i)19-s + 3.58·21-s + 0.202·23-s + (4.31 + 13.2i)25-s + (−8.88 − 6.45i)27-s + (−1.81 + 5.57i)29-s + (1.85 − 1.34i)31-s + ⋯
L(s)  = 1  + (0.551 + 1.69i)3-s + (−1.57 − 1.14i)5-s + (0.135 − 0.416i)7-s + (−1.76 + 1.28i)9-s + (−0.915 + 0.401i)11-s + (0.224 − 0.163i)13-s + (1.07 − 3.30i)15-s + (−1.48 − 1.07i)17-s + (−0.401 − 1.23i)19-s + 0.781·21-s + 0.0422·23-s + (0.863 + 2.65i)25-s + (−1.71 − 1.24i)27-s + (−0.336 + 1.03i)29-s + (0.332 − 0.241i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 + 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(572\)    =    \(2^{2} \cdot 11 \cdot 13\)
Sign: $-0.525 + 0.850i$
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: \((2,\ 572,\ (\ :1/2),\ -0.525 + 0.850i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0541183 - 0.0969921i\)
\(L(\frac12)\) \(\approx\) \(0.0541183 - 0.0969921i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.03 - 1.33i)T \)
13 \( 1 + (-0.809 + 0.587i)T \)
good3 \( 1 + (-0.955 - 2.93i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (3.52 + 2.55i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (-0.357 + 1.10i)T + (-5.66 - 4.11i)T^{2} \)
17 \( 1 + (6.11 + 4.44i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.75 + 5.38i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 0.202T + 23T^{2} \)
29 \( 1 + (1.81 - 5.57i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-1.85 + 1.34i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (0.723 - 2.22i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (0.719 + 2.21i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 3.95T + 43T^{2} \)
47 \( 1 + (0.684 + 2.10i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (0.0852 - 0.0619i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (2.05 - 6.31i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (8.65 + 6.28i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 9.05T + 67T^{2} \)
71 \( 1 + (-0.100 - 0.0733i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (0.654 - 2.01i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-10.1 + 7.40i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-6.63 - 4.81i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 14.0T + 89T^{2} \)
97 \( 1 + (-2.25 + 1.63i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.59692733252253031579197899896, −9.261235239282541751303467104049, −8.881537640312318158892022308655, −8.062425582800823327124115377508, −7.16484500138125899150666145473, −5.01349666169076020002587322991, −4.74846350131749428368733263673, −3.95918294963886253614652533842, −2.86026634330444913193795733151, −0.05534047742692327855682340579, 2.06068374720281571996431039431, 3.02273498123380694521338256625, 4.07587735814108893453637123549, 6.05823760834644637567166572821, 6.67085171929458116539641589007, 7.62945870766156799596959762481, 8.132738033398559607591136213416, 8.678538803318470504232389277490, 10.49358775441433510194758796101, 11.22000503140698271145460107571

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