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Properties

Label 2-572-13.4-c1-0-6
Degree $2$
Conductor $572$
Sign $0.794 + 0.607i$
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  + (−1.07 + 1.86i)3-s − 3.60i·5-s + (−3.54 + 2.04i)7-s + (−0.819 − 1.41i)9-s + (0.866 + 0.5i)11-s + (3.34 − 1.34i)13-s + (6.72 + 3.88i)15-s + (−1.83 − 3.17i)17-s + (5.54 − 3.20i)19-s − 8.81i·21-s + (4.23 − 7.32i)23-s − 7.98·25-s − 2.93·27-s + (4.18 − 7.24i)29-s + 6.40i·31-s + ⋯
L(s)  = 1  + (−0.621 + 1.07i)3-s − 1.61i·5-s + (−1.33 + 0.773i)7-s + (−0.273 − 0.473i)9-s + (0.261 + 0.150i)11-s + (0.928 − 0.372i)13-s + (1.73 + 1.00i)15-s + (−0.444 − 0.769i)17-s + (1.27 − 0.734i)19-s − 1.92i·21-s + (0.882 − 1.52i)23-s − 1.59·25-s − 0.564·27-s + (0.776 − 1.34i)29-s + 1.15i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.794 + 0.607i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.794 + 0.607i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(572\)    =    \(2^{2} \cdot 11 \cdot 13\)
Sign: $0.794 + 0.607i$
Analytic conductor: \(4.56744\)
Root analytic conductor: \(2.13715\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{572} (485, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 572,\ (\ :1/2),\ 0.794 + 0.607i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.894710 - 0.303019i\)
\(L(\frac12)\) \(\approx\) \(0.894710 - 0.303019i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.866 - 0.5i)T \)
13 \( 1 + (-3.34 + 1.34i)T \)
good3 \( 1 + (1.07 - 1.86i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + 3.60iT - 5T^{2} \)
7 \( 1 + (3.54 - 2.04i)T + (3.5 - 6.06i)T^{2} \)
17 \( 1 + (1.83 + 3.17i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-5.54 + 3.20i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.23 + 7.32i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.18 + 7.24i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 6.40iT - 31T^{2} \)
37 \( 1 + (-2.69 - 1.55i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.62 - 2.09i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.72 + 2.98i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 10.5iT - 47T^{2} \)
53 \( 1 + 0.556T + 53T^{2} \)
59 \( 1 + (-0.792 + 0.457i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.93 - 6.81i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (9.87 + 5.69i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (8.21 - 4.74i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 1.80iT - 73T^{2} \)
79 \( 1 - 1.54T + 79T^{2} \)
83 \( 1 - 4.44iT - 83T^{2} \)
89 \( 1 + (-1.19 - 0.687i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.02 - 3.47i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.48655648389612980582426900695, −9.614198210159056983395118738894, −9.100492478232140674793615070316, −8.428965498185369471966432724902, −6.78734897111784258763095370806, −5.75228179464839056443434014743, −5.03219036054003990000514435735, −4.25604740705353904304252267863, −2.92608534494614431377846689495, −0.66034856767088682106915213101, 1.32241980809882657975545412448, 3.10118201636869915620950686602, 3.77016541616256016844969972176, 5.95856070449464647566017909697, 6.32682050045565047137349465717, 7.13687303380037395270123324530, 7.60394033799049582571215300359, 9.268139400246410278994041599388, 10.10396613915340265832503288154, 11.02046043581956283660233139905

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