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Properties

Label 2-572-11.4-c1-0-1
Degree $2$
Conductor $572$
Sign $-0.845 - 0.533i$
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  + (−0.427 + 1.31i)3-s + (−1.65 + 1.20i)5-s + (0.287 + 0.884i)7-s + (0.879 + 0.639i)9-s + (3.31 − 0.137i)11-s + (−0.809 − 0.587i)13-s + (−0.876 − 2.69i)15-s + (−3.76 + 2.73i)17-s + (−1.26 + 3.89i)19-s − 1.28·21-s − 6.10·23-s + (−0.244 + 0.751i)25-s + (−4.57 + 3.32i)27-s + (−2.15 − 6.64i)29-s + (−5.80 − 4.21i)31-s + ⋯
L(s)  = 1  + (−0.246 + 0.759i)3-s + (−0.742 + 0.539i)5-s + (0.108 + 0.334i)7-s + (0.293 + 0.213i)9-s + (0.999 − 0.0415i)11-s + (−0.224 − 0.163i)13-s + (−0.226 − 0.696i)15-s + (−0.913 + 0.663i)17-s + (−0.290 + 0.892i)19-s − 0.280·21-s − 1.27·23-s + (−0.0488 + 0.150i)25-s + (−0.880 + 0.639i)27-s + (−0.401 − 1.23i)29-s + (−1.04 − 0.757i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.845 - 0.533i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.845 - 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(572\)    =    \(2^{2} \cdot 11 \cdot 13\)
Sign: $-0.845 - 0.533i$
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: \((2,\ 572,\ (\ :1/2),\ -0.845 - 0.533i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.252798 + 0.874193i\)
\(L(\frac12)\) \(\approx\) \(0.252798 + 0.874193i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.31 + 0.137i)T \)
13 \( 1 + (0.809 + 0.587i)T \)
good3 \( 1 + (0.427 - 1.31i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 + (1.65 - 1.20i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (-0.287 - 0.884i)T + (-5.66 + 4.11i)T^{2} \)
17 \( 1 + (3.76 - 2.73i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (1.26 - 3.89i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 6.10T + 23T^{2} \)
29 \( 1 + (2.15 + 6.64i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (5.80 + 4.21i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-1.69 - 5.22i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (3.56 - 10.9i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 5.01T + 43T^{2} \)
47 \( 1 + (-2.43 + 7.48i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-11.5 - 8.36i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (4.17 + 12.8i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-2.00 + 1.45i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 4.44T + 67T^{2} \)
71 \( 1 + (3.28 - 2.38i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-3.68 - 11.3i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-11.0 - 8.05i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (11.2 - 8.13i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 11.8T + 89T^{2} \)
97 \( 1 + (-3.86 - 2.80i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.20018071298983297198377851125, −10.18887009071468584638332123450, −9.563692923499857914770471923766, −8.413009904273416642736150667626, −7.61083491216144051587072192276, −6.51686719416792149975392395137, −5.57732291861880805673610195610, −4.15106699288634517853983936528, −3.84633218864037306345469407852, −2.05983102501156844014151452050, 0.53180331935903202936294219547, 1.97010222651520140735095025840, 3.83767924066454596839956832842, 4.53859808900403608711839019095, 5.91199009196696852624268437174, 7.10811882094046480909064614735, 7.30674371230825024277850122403, 8.722294178242517772943518140612, 9.220032580708304036120751166962, 10.54162472095552520247472997635

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