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Properties

Label 2-572-13.3-c1-0-1
Degree $2$
Conductor $572$
Sign $0.872 - 0.488i$
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  + (−1.23 − 2.13i)3-s − 2.05·5-s + (−1.43 + 2.48i)7-s + (−1.52 + 2.64i)9-s + (0.5 + 0.866i)11-s + (2.5 + 2.59i)13-s + (2.52 + 4.37i)15-s + (2.82 − 4.89i)17-s + (−1.73 + 2.99i)19-s + 7.05·21-s + (4.28 + 7.42i)23-s − 0.780·25-s + 0.133·27-s + (−3.39 − 5.87i)29-s + 10.8·31-s + ⋯
L(s)  = 1  + (−0.710 − 1.23i)3-s − 0.918·5-s + (−0.541 + 0.938i)7-s + (−0.509 + 0.881i)9-s + (0.150 + 0.261i)11-s + (0.693 + 0.720i)13-s + (0.652 + 1.13i)15-s + (0.684 − 1.18i)17-s + (−0.396 + 0.687i)19-s + 1.53·21-s + (0.893 + 1.54i)23-s − 0.156·25-s + 0.0256·27-s + (−0.630 − 1.09i)29-s + 1.94·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 - 0.488i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.872 - 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(572\)    =    \(2^{2} \cdot 11 \cdot 13\)
Sign: $0.872 - 0.488i$
Analytic conductor: \(4.56744\)
Root analytic conductor: \(2.13715\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{572} (133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 572,\ (\ :1/2),\ 0.872 - 0.488i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.712065 + 0.185912i\)
\(L(\frac12)\) \(\approx\) \(0.712065 + 0.185912i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (-2.5 - 2.59i)T \)
good3 \( 1 + (1.23 + 2.13i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + 2.05T + 5T^{2} \)
7 \( 1 + (1.43 - 2.48i)T + (-3.5 - 6.06i)T^{2} \)
17 \( 1 + (-2.82 + 4.89i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.73 - 2.99i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.28 - 7.42i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.39 + 5.87i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 10.8T + 31T^{2} \)
37 \( 1 + (1.12 + 1.94i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.82 - 4.89i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.71 - 8.16i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 5.32T + 47T^{2} \)
53 \( 1 - 10.9T + 53T^{2} \)
59 \( 1 + (6.39 - 11.0i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.71 - 9.90i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.429 + 0.744i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (3.81 - 6.60i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 4.37T + 73T^{2} \)
79 \( 1 + 16.4T + 79T^{2} \)
83 \( 1 - 7.97T + 83T^{2} \)
89 \( 1 + (-0.0811 - 0.140i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.42 - 9.38i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.34614988963358320764071666787, −9.861292936107111647687242215039, −8.977459079147982553235416863995, −7.85485436067257818286615937912, −7.25342340045002383056416827643, −6.26297841522057564459277294577, −5.60886230080885425077968322097, −4.16591911226042070490529088578, −2.79369619696306451321851505813, −1.23175758434516573003712055196, 0.53697340599067567078445436559, 3.34601554392530329748395855602, 3.97739492828267563469898087458, 4.85634244555337405643000134398, 6.01207414443416946951908129223, 6.95176240290876145793135573932, 8.134276407307926814339698189407, 8.936952248871635759018910521484, 10.30572600220176636354476278672, 10.48385434021865345443014304084

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