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Properties

Label 2-572-13.4-c1-0-7
Degree $2$
Conductor $572$
Sign $0.998 + 0.0512i$
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  + (0.716 − 1.24i)3-s + 2.80i·5-s + (4.48 − 2.59i)7-s + (0.473 + 0.820i)9-s + (−0.866 − 0.5i)11-s + (−1.99 + 3.00i)13-s + (3.47 + 2.00i)15-s + (1.08 + 1.87i)17-s + (−1.69 + 0.978i)19-s − 7.42i·21-s + (2.85 − 4.95i)23-s − 2.85·25-s + 5.65·27-s + (−2.78 + 4.82i)29-s − 4.69i·31-s + ⋯
L(s)  = 1  + (0.413 − 0.716i)3-s + 1.25i·5-s + (1.69 − 0.979i)7-s + (0.157 + 0.273i)9-s + (−0.261 − 0.150i)11-s + (−0.554 + 0.832i)13-s + (0.897 + 0.518i)15-s + (0.263 + 0.455i)17-s + (−0.388 + 0.224i)19-s − 1.62i·21-s + (0.595 − 1.03i)23-s − 0.570·25-s + 1.08·27-s + (−0.516 + 0.895i)29-s − 0.843i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0512i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0512i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(572\)    =    \(2^{2} \cdot 11 \cdot 13\)
Sign: $0.998 + 0.0512i$
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: \((2,\ 572,\ (\ :1/2),\ 0.998 + 0.0512i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.96260 - 0.0503381i\)
\(L(\frac12)\) \(\approx\) \(1.96260 - 0.0503381i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.99 - 3.00i)T \)
good3 \( 1 + (-0.716 + 1.24i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 - 2.80iT - 5T^{2} \)
7 \( 1 + (-4.48 + 2.59i)T + (3.5 - 6.06i)T^{2} \)
17 \( 1 + (-1.08 - 1.87i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.69 - 0.978i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.85 + 4.95i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.78 - 4.82i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 4.69iT - 31T^{2} \)
37 \( 1 + (-4.30 - 2.48i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.64 - 2.10i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.38 + 9.32i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 4.57iT - 47T^{2} \)
53 \( 1 - 1.56T + 53T^{2} \)
59 \( 1 + (11.3 - 6.57i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.29 + 10.9i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.44 - 3.71i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (7.51 - 4.33i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 2.25iT - 73T^{2} \)
79 \( 1 + 6.15T + 79T^{2} \)
83 \( 1 - 6.08iT - 83T^{2} \)
89 \( 1 + (-2.80 - 1.62i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.75 - 2.16i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.78405288126361787163107707668, −10.18836126742547813754814102642, −8.666863691155571680438650182427, −7.82413770635412690378337964428, −7.28160835650238585785480824424, −6.57314768539417091066698546188, −5.02134252289188354229323821034, −4.05549160304217791854878020247, −2.55117685795178675542099607726, −1.58504793161152517728578932605, 1.37453734391025090227164435372, 2.83739347124055203584472441355, 4.45332201458458887477081823225, 4.95007471969474171748023818378, 5.73534003846362795803414745391, 7.58611334005272130622613063561, 8.226055665830813781700199493014, 9.066791360503406304319443442968, 9.542388209229146540342409119321, 10.74484286653066490289794995950

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