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Properties

Label 2-572-572.571-c1-0-69
Degree $2$
Conductor $572$
Sign $-0.981 - 0.193i$
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  + (−1.09 + 0.897i)2-s − 3.43i·3-s + (0.387 − 1.96i)4-s + (3.08 + 3.74i)6-s − 0.803i·7-s + (1.33 + 2.49i)8-s − 8.76·9-s − 3.31i·11-s + (−6.73 − 1.33i)12-s − 3.60·13-s + (0.721 + 0.878i)14-s + (−3.69 − 1.52i)16-s + (9.58 − 7.87i)18-s + 5.18i·19-s − 2.75·21-s + (2.97 + 3.62i)22-s + ⋯
L(s)  = 1  + (−0.772 + 0.634i)2-s − 1.98i·3-s + (0.193 − 0.981i)4-s + (1.25 + 1.53i)6-s − 0.303i·7-s + (0.472 + 0.881i)8-s − 2.92·9-s − 1.00i·11-s + (−1.94 − 0.384i)12-s − 1.00·13-s + (0.192 + 0.234i)14-s + (−0.924 − 0.380i)16-s + (2.25 − 1.85i)18-s + 1.18i·19-s − 0.601·21-s + (0.634 + 0.772i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 - 0.193i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.981 - 0.193i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(572\)    =    \(2^{2} \cdot 11 \cdot 13\)
Sign: $-0.981 - 0.193i$
Analytic conductor: \(4.56744\)
Root analytic conductor: \(2.13715\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{572} (571, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 572,\ (\ :1/2),\ -0.981 - 0.193i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0490322 + 0.500909i\)
\(L(\frac12)\) \(\approx\) \(0.0490322 + 0.500909i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.09 - 0.897i)T \)
11 \( 1 + 3.31iT \)
13 \( 1 + 3.60T \)
good3 \( 1 + 3.43iT - 3T^{2} \)
5 \( 1 - 5T^{2} \)
7 \( 1 + 0.803iT - 7T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 - 5.18iT - 19T^{2} \)
23 \( 1 + 6.19iT - 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + 10.3T + 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 14.3T + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 6.10T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 17.5iT - 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−10.28420928206217505915181839382, −8.917099968942698136448803051402, −8.258626751286913432478673272914, −7.61112520349852505883150012907, −6.75412991166949300197439491369, −6.17388770193757979576990521635, −5.17162554955619176014114604540, −2.86444777065539320538359614879, −1.62234578428158458820098090406, −0.35696472529627918496203528366, 2.46729648251359805965006727976, 3.41692265147147665429005651494, 4.56731605902093472418766739367, 5.18953806567214254126075263882, 6.86432430437416861510233962384, 8.065934406976296981952880917177, 9.110951040481267917895798726607, 9.497353652926852143571175480587, 10.17632793161900537010014871065, 10.94867215071101051126693457802

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