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Properties

Label 2-572-11.9-c1-0-2
Degree $2$
Conductor $572$
Sign $0.331 - 0.943i$
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  + (0.445 − 0.323i)3-s + (0.710 + 2.18i)5-s + (3.17 + 2.30i)7-s + (−0.833 + 2.56i)9-s + (−3.07 − 1.24i)11-s + (0.309 − 0.951i)13-s + (1.02 + 0.744i)15-s + (0.370 + 1.13i)17-s + (−5.34 + 3.88i)19-s + 2.16·21-s − 4.07·23-s + (−0.225 + 0.164i)25-s + (0.970 + 2.98i)27-s + (0.649 + 0.471i)29-s + (2.37 − 7.31i)31-s + ⋯
L(s)  = 1  + (0.257 − 0.187i)3-s + (0.317 + 0.977i)5-s + (1.19 + 0.871i)7-s + (−0.277 + 0.854i)9-s + (−0.927 − 0.374i)11-s + (0.0857 − 0.263i)13-s + (0.264 + 0.192i)15-s + (0.0898 + 0.276i)17-s + (−1.22 + 0.890i)19-s + 0.471·21-s − 0.850·23-s + (−0.0451 + 0.0328i)25-s + (0.186 + 0.574i)27-s + (0.120 + 0.0876i)29-s + (0.426 − 1.31i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.331 - 0.943i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.331 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(572\)    =    \(2^{2} \cdot 11 \cdot 13\)
Sign: $0.331 - 0.943i$
Analytic conductor: \(4.56744\)
Root analytic conductor: \(2.13715\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{572} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 572,\ (\ :1/2),\ 0.331 - 0.943i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.36309 + 0.965726i\)
\(L(\frac12)\) \(\approx\) \(1.36309 + 0.965726i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.07 + 1.24i)T \)
13 \( 1 + (-0.309 + 0.951i)T \)
good3 \( 1 + (-0.445 + 0.323i)T + (0.927 - 2.85i)T^{2} \)
5 \( 1 + (-0.710 - 2.18i)T + (-4.04 + 2.93i)T^{2} \)
7 \( 1 + (-3.17 - 2.30i)T + (2.16 + 6.65i)T^{2} \)
17 \( 1 + (-0.370 - 1.13i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (5.34 - 3.88i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 4.07T + 23T^{2} \)
29 \( 1 + (-0.649 - 0.471i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-2.37 + 7.31i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-5.68 - 4.12i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-3.10 + 2.25i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 12.3T + 43T^{2} \)
47 \( 1 + (-9.33 + 6.77i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (2.21 - 6.82i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (3.45 + 2.50i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-1.52 - 4.69i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 - 4.27T + 67T^{2} \)
71 \( 1 + (3.32 + 10.2i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (4.82 + 3.50i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-2.82 + 8.69i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (0.235 + 0.724i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 1.79T + 89T^{2} \)
97 \( 1 + (2.14 - 6.59i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.66049116998554557795147958860, −10.46532251998340725142120312019, −8.968216518233081942355093211464, −7.992660718759005551336003722860, −7.76275737202106670690245169305, −6.13903993399391526355739577679, −5.58289382318098365319368744893, −4.33298865815151482707843706595, −2.65821414700322599507913530357, −2.10759658775407998160021818169, 0.959196513201054819678335032664, 2.45087030849985427802961478822, 4.18617426741825714454145328968, 4.72059941497367870997725040493, 5.83530337341737810175729878828, 7.10294170337029462917124108153, 8.066989970337542515063320963760, 8.772720225810602871730112217094, 9.579238965117021678213449373568, 10.60988708909814906898043622057

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