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Properties

Label 2-572-11.3-c1-0-2
Degree $2$
Conductor $572$
Sign $0.809 - 0.587i$
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  + (0.288 + 0.886i)3-s + (−2.56 − 1.86i)5-s + (−0.780 + 2.40i)7-s + (1.72 − 1.25i)9-s + (3.31 − 0.0787i)11-s + (0.809 − 0.587i)13-s + (0.914 − 2.81i)15-s + (2.29 + 1.66i)17-s + (2.68 + 8.26i)19-s − 2.35·21-s + 5.07·23-s + (1.56 + 4.83i)25-s + (3.86 + 2.81i)27-s + (−0.125 + 0.387i)29-s + (−3.27 + 2.38i)31-s + ⋯
L(s)  = 1  + (0.166 + 0.511i)3-s + (−1.14 − 0.834i)5-s + (−0.294 + 0.907i)7-s + (0.574 − 0.417i)9-s + (0.999 − 0.0237i)11-s + (0.224 − 0.163i)13-s + (0.236 − 0.726i)15-s + (0.556 + 0.404i)17-s + (0.616 + 1.89i)19-s − 0.513·21-s + 1.05·23-s + (0.313 + 0.966i)25-s + (0.744 + 0.541i)27-s + (−0.0233 + 0.0719i)29-s + (−0.588 + 0.427i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.809 - 0.587i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.809 - 0.587i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(572\)    =    \(2^{2} \cdot 11 \cdot 13\)
Sign: $0.809 - 0.587i$
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: \((2,\ 572,\ (\ :1/2),\ 0.809 - 0.587i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.30563 + 0.424216i\)
\(L(\frac12)\) \(\approx\) \(1.30563 + 0.424216i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.31 + 0.0787i)T \)
13 \( 1 + (-0.809 + 0.587i)T \)
good3 \( 1 + (-0.288 - 0.886i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (2.56 + 1.86i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (0.780 - 2.40i)T + (-5.66 - 4.11i)T^{2} \)
17 \( 1 + (-2.29 - 1.66i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-2.68 - 8.26i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 5.07T + 23T^{2} \)
29 \( 1 + (0.125 - 0.387i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (3.27 - 2.38i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-2.80 + 8.63i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-0.885 - 2.72i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 4.20T + 43T^{2} \)
47 \( 1 + (-0.403 - 1.24i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-0.446 + 0.324i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-3.34 + 10.2i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (1.63 + 1.18i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 7.10T + 67T^{2} \)
71 \( 1 + (-1.10 - 0.805i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (2.29 - 7.06i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-11.4 + 8.32i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (0.527 + 0.383i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 9.85T + 89T^{2} \)
97 \( 1 + (15.3 - 11.1i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.84846237529488577858829427652, −9.702953437937204640485711333838, −9.091720640599699841878293434986, −8.330878966556496491263358808916, −7.41781842783227994379045421931, −6.17341433393323342358423643513, −5.14399994552061844447768471740, −3.96044163952308804232723937324, −3.44774009042537370505603039226, −1.29727153209905527560966275640, 0.981448639913083216887231474712, 2.88090721856386423673697436824, 3.84397363890150811351938263319, 4.80257890950288139009967552044, 6.60982068253424051611369742787, 7.15471334062370474685803761913, 7.58848944922719009940427170795, 8.821886868877625762073519533800, 9.834511580693077239374595999656, 10.84558640570788923880506662834

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