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Properties

Label 2-572-572.263-c1-0-23
Degree $2$
Conductor $572$
Sign $-0.136 - 0.990i$
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  + (−0.499 + 1.32i)2-s + (−2.08 + 1.20i)3-s + (−1.50 − 1.32i)4-s − 2.36·5-s + (−0.551 − 3.35i)6-s + (1.42 − 2.46i)7-s + (2.49 − 1.32i)8-s + (1.39 − 2.40i)9-s + (1.17 − 3.12i)10-s + (3.21 − 0.819i)11-s + (4.71 + 0.946i)12-s + (−2.34 + 2.74i)13-s + (2.54 + 3.11i)14-s + (4.92 − 2.84i)15-s + (0.508 + 3.96i)16-s + (−1.53 − 0.886i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.935i)2-s + (−1.20 + 0.694i)3-s + (−0.750 − 0.660i)4-s − 1.05·5-s + (−0.224 − 1.36i)6-s + (0.537 − 0.931i)7-s + (0.883 − 0.469i)8-s + (0.463 − 0.802i)9-s + (0.373 − 0.988i)10-s + (0.968 − 0.247i)11-s + (1.36 + 0.273i)12-s + (−0.649 + 0.760i)13-s + (0.681 + 0.831i)14-s + (1.27 − 0.733i)15-s + (0.127 + 0.991i)16-s + (−0.372 − 0.215i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.136 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.136 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(572\)    =    \(2^{2} \cdot 11 \cdot 13\)
Sign: $-0.136 - 0.990i$
Analytic conductor: \(4.56744\)
Root analytic conductor: \(2.13715\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{572} (263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 572,\ (\ :1/2),\ -0.136 - 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.371269 + 0.426004i\)
\(L(\frac12)\) \(\approx\) \(0.371269 + 0.426004i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.499 - 1.32i)T \)
11 \( 1 + (-3.21 + 0.819i)T \)
13 \( 1 + (2.34 - 2.74i)T \)
good3 \( 1 + (2.08 - 1.20i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + 2.36T + 5T^{2} \)
7 \( 1 + (-1.42 + 2.46i)T + (-3.5 - 6.06i)T^{2} \)
17 \( 1 + (1.53 + 0.886i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.0598 - 0.103i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-5.93 + 3.42i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.88 - 1.66i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 6.06iT - 31T^{2} \)
37 \( 1 + (-2.26 - 3.92i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-7.34 + 4.23i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.01 - 6.94i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 2.87iT - 47T^{2} \)
53 \( 1 - 5.47T + 53T^{2} \)
59 \( 1 + (-12.7 - 7.36i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.95 - 1.13i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.27 + 4.20i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.24 + 0.716i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 15.8iT - 73T^{2} \)
79 \( 1 - 13.1T + 79T^{2} \)
83 \( 1 + 9.19T + 83T^{2} \)
89 \( 1 + (-3.41 - 5.90i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.294 - 0.510i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.05193194043867422875925377916, −10.14209677038719911972924022308, −9.183817311185948764821194056242, −8.245042503446753635549515287558, −7.11413234551084162733181447205, −6.70495136069604707713143359479, −5.35861004284051005296814925993, −4.48793224990731049893717509785, −4.01311738177327962356348296189, −0.856287190089151949197204663140, 0.66259443724711326550250692261, 2.14282963236203785224656198528, 3.69707962483856668156104809582, 4.84594671916712387098681497602, 5.75020896126781884464810386069, 7.11576269627982218875822760672, 7.81027885760267916987212417604, 8.797620936148685602614129673483, 9.703180145439699930233356460926, 11.04243016393132669057579496053

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