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Properties

Label 2-572-11.5-c1-0-8
Degree $2$
Conductor $572$
Sign $-0.490 + 0.871i$
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  + (−1.94 − 1.41i)3-s + (0.879 − 2.70i)5-s + (3.92 − 2.84i)7-s + (0.855 + 2.63i)9-s + (2.96 + 1.49i)11-s + (−0.309 − 0.951i)13-s + (−5.53 + 4.02i)15-s + (0.656 − 2.01i)17-s + (2.27 + 1.65i)19-s − 11.6·21-s + 6.11·23-s + (−2.51 − 1.82i)25-s + (−0.171 + 0.528i)27-s + (−6.73 + 4.89i)29-s + (−2.46 − 7.58i)31-s + ⋯
L(s)  = 1  + (−1.12 − 0.815i)3-s + (0.393 − 1.21i)5-s + (1.48 − 1.07i)7-s + (0.285 + 0.877i)9-s + (0.892 + 0.450i)11-s + (−0.0857 − 0.263i)13-s + (−1.42 + 1.03i)15-s + (0.159 − 0.489i)17-s + (0.521 + 0.379i)19-s − 2.54·21-s + 1.27·23-s + (−0.502 − 0.365i)25-s + (−0.0330 + 0.101i)27-s + (−1.25 + 0.908i)29-s + (−0.442 − 1.36i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.490 + 0.871i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.490 + 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(572\)    =    \(2^{2} \cdot 11 \cdot 13\)
Sign: $-0.490 + 0.871i$
Analytic conductor: \(4.56744\)
Root analytic conductor: \(2.13715\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{572} (313, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 572,\ (\ :1/2),\ -0.490 + 0.871i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.656431 - 1.12302i\)
\(L(\frac12)\) \(\approx\) \(0.656431 - 1.12302i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.96 - 1.49i)T \)
13 \( 1 + (0.309 + 0.951i)T \)
good3 \( 1 + (1.94 + 1.41i)T + (0.927 + 2.85i)T^{2} \)
5 \( 1 + (-0.879 + 2.70i)T + (-4.04 - 2.93i)T^{2} \)
7 \( 1 + (-3.92 + 2.84i)T + (2.16 - 6.65i)T^{2} \)
17 \( 1 + (-0.656 + 2.01i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-2.27 - 1.65i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 6.11T + 23T^{2} \)
29 \( 1 + (6.73 - 4.89i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (2.46 + 7.58i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (4.56 - 3.31i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (6.95 + 5.05i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 4.16T + 43T^{2} \)
47 \( 1 + (-0.986 - 0.716i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-3.81 - 11.7i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (6.92 - 5.03i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-0.363 + 1.11i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 9.43T + 67T^{2} \)
71 \( 1 + (1.28 - 3.96i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (11.4 - 8.29i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-3.65 - 11.2i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (4.31 - 13.2i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 2.42T + 89T^{2} \)
97 \( 1 + (4.02 + 12.3i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.76388729573982429646290711113, −9.538386406203725585874595529200, −8.640796607708294170411776484867, −7.46235967677070465523279533462, −7.04201637579158757396317463466, −5.56509879086678765425296915608, −5.10252183201569797992249818471, −4.08914129767017616192826378057, −1.56470702919780770863425512807, −1.00560930338757669987334078012, 1.85627637060447205548221729600, 3.35938709815064859654311225367, 4.74104992691009624609101597880, 5.47551904690519109032636249020, 6.24953841188236757215784128794, 7.24760507620088443915680834180, 8.570694147259466213619946470941, 9.389163369288214482268001110348, 10.44572268684041259887003351369, 11.17168024217786108734238558875

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