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Properties

Label 2-572-572.51-c1-0-78
Degree $2$
Conductor $572$
Sign $-0.722 - 0.691i$
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  + (0.831 − 1.14i)2-s + (−0.618 − 1.90i)4-s + (−3.93 + 1.27i)7-s + (−2.68 − 0.874i)8-s + (−2.42 − 1.76i)9-s + (−1.22 + 3.08i)11-s + (−2.11 + 2.91i)13-s + (−1.80 + 5.56i)14-s + (−3.23 + 2.35i)16-s + (−4.81 − 6.62i)17-s + (−4.03 + 1.31i)18-s + (6.69 + 2.17i)19-s + (2.50 + 3.96i)22-s + (1.54 − 4.75i)25-s + (1.57 + 4.84i)26-s + ⋯
L(s)  = 1  + (0.587 − 0.809i)2-s + (−0.309 − 0.951i)4-s + (−1.48 + 0.483i)7-s + (−0.951 − 0.309i)8-s + (−0.809 − 0.587i)9-s + (−0.370 + 0.928i)11-s + (−0.587 + 0.809i)13-s + (−0.483 + 1.48i)14-s + (−0.809 + 0.587i)16-s + (−1.16 − 1.60i)17-s + (−0.951 + 0.309i)18-s + (1.53 + 0.498i)19-s + (0.533 + 0.845i)22-s + (0.309 − 0.951i)25-s + (0.309 + 0.951i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.722 - 0.691i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.722 - 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(572\)    =    \(2^{2} \cdot 11 \cdot 13\)
Sign: $-0.722 - 0.691i$
Analytic conductor: \(4.56744\)
Root analytic conductor: \(2.13715\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{572} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 572,\ (\ :1/2),\ -0.722 - 0.691i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.110281 + 0.274707i\)
\(L(\frac12)\) \(\approx\) \(0.110281 + 0.274707i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.831 + 1.14i)T \)
11 \( 1 + (1.22 - 3.08i)T \)
13 \( 1 + (2.11 - 2.91i)T \)
good3 \( 1 + (2.42 + 1.76i)T^{2} \)
5 \( 1 + (-1.54 + 4.75i)T^{2} \)
7 \( 1 + (3.93 - 1.27i)T + (5.66 - 4.11i)T^{2} \)
17 \( 1 + (4.81 + 6.62i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (-6.69 - 2.17i)T + (15.3 + 11.1i)T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + (-5.11 + 1.66i)T + (23.4 - 17.0i)T^{2} \)
31 \( 1 + (8.89 + 6.46i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + (1.03 - 3.19i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (10.5 + 7.69i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-1.04 - 3.21i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-5.97 - 8.22i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 + 5.09T + 67T^{2} \)
71 \( 1 + (6.64 - 4.83i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (6.75 + 9.29i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 + (-29.9 - 92.2i)T^{2} \)
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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

−9.980454654797931040304450040824, −9.534599639231698366499757042672, −8.962631306649534652981170938366, −7.19315663693202518323461265107, −6.39530983396161903828048692836, −5.42902661586226764285731001397, −4.38389411152472027442120608329, −3.09094486771455133157473589720, −2.41060956773858906310958411184, −0.12658864899021276974019234635, 2.96342307758594164809348958614, 3.48791286138285173401839461981, 5.03804086543620193552014677446, 5.80428741497852363371527386635, 6.68827893466174052285405479250, 7.56202173332484118495033423073, 8.518962744390484010313485822584, 9.285686254100719763926508199330, 10.50292209072931026193992600673, 11.21841638241264859326752026271

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