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Properties

Label 2-572-11.4-c1-0-5
Degree $2$
Conductor $572$
Sign $0.865 - 0.500i$
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  + (−0.527 + 1.62i)3-s + (1.55 − 1.13i)5-s + (−0.0663 − 0.204i)7-s + (0.0710 + 0.0516i)9-s + (1.41 − 2.99i)11-s + (0.809 + 0.587i)13-s + (1.01 + 3.12i)15-s + (5.55 − 4.03i)17-s + (−1.19 + 3.68i)19-s + 0.366·21-s + 3.40·23-s + (−0.400 + 1.23i)25-s + (−4.26 + 3.09i)27-s + (−0.0803 − 0.247i)29-s + (1.10 + 0.800i)31-s + ⋯
L(s)  = 1  + (−0.304 + 0.937i)3-s + (0.696 − 0.506i)5-s + (−0.0250 − 0.0772i)7-s + (0.0236 + 0.0172i)9-s + (0.428 − 0.903i)11-s + (0.224 + 0.163i)13-s + (0.262 + 0.806i)15-s + (1.34 − 0.978i)17-s + (−0.274 + 0.845i)19-s + 0.0800·21-s + 0.709·23-s + (−0.0800 + 0.246i)25-s + (−0.820 + 0.596i)27-s + (−0.0149 − 0.0459i)29-s + (0.197 + 0.143i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.865 - 0.500i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.865 - 0.500i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(572\)    =    \(2^{2} \cdot 11 \cdot 13\)
Sign: $0.865 - 0.500i$
Analytic conductor: \(4.56744\)
Root analytic conductor: \(2.13715\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{572} (521, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 572,\ (\ :1/2),\ 0.865 - 0.500i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.55205 + 0.416088i\)
\(L(\frac12)\) \(\approx\) \(1.55205 + 0.416088i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.41 + 2.99i)T \)
13 \( 1 + (-0.809 - 0.587i)T \)
good3 \( 1 + (0.527 - 1.62i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 + (-1.55 + 1.13i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (0.0663 + 0.204i)T + (-5.66 + 4.11i)T^{2} \)
17 \( 1 + (-5.55 + 4.03i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (1.19 - 3.68i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 3.40T + 23T^{2} \)
29 \( 1 + (0.0803 + 0.247i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-1.10 - 0.800i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (0.532 + 1.63i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (0.417 - 1.28i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 1.40T + 43T^{2} \)
47 \( 1 + (3.71 - 11.4i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-5.54 - 4.02i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (1.14 + 3.53i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-8.20 + 5.96i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 8.18T + 67T^{2} \)
71 \( 1 + (7.53 - 5.47i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-1.76 - 5.42i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (2.80 + 2.04i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-8.07 + 5.86i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 6.53T + 89T^{2} \)
97 \( 1 + (10.9 + 7.96i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.71917383639320580397956838057, −9.823343008656169315326451450970, −9.341535586257244418723026384566, −8.356444377650275757930229790707, −7.20781794410041216816101240193, −5.87054559361106219953744369468, −5.33837483229848881675517451565, −4.25096145138240519311322892756, −3.17138126172781664617053805305, −1.31462545677096405201740822624, 1.28434470650543923367010660226, 2.43047838447094639776411947768, 3.90942873444309187116003853432, 5.35071624891410358839452172553, 6.31141460391758337607523467224, 6.91768936001739217113392996048, 7.76808311326381405781001097970, 8.922923873747349225367285198224, 9.954380588576898332368282147944, 10.52098169597237644295684341952

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