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Properties

Label 2-572-13.3-c1-0-0
Degree $2$
Conductor $572$
Sign $-0.476 - 0.879i$
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  + (−0.542 − 0.939i)3-s − 1.28·5-s + (−1.24 + 2.16i)7-s + (0.911 − 1.57i)9-s + (−0.5 − 0.866i)11-s + (−2.63 + 2.46i)13-s + (0.695 + 1.20i)15-s + (−3.07 + 5.31i)17-s + (0.187 − 0.323i)19-s + 2.70·21-s + (0.919 + 1.59i)23-s − 3.35·25-s − 5.23·27-s + (4.13 + 7.15i)29-s − 6.44·31-s + ⋯
L(s)  = 1  + (−0.313 − 0.542i)3-s − 0.573·5-s + (−0.471 + 0.816i)7-s + (0.303 − 0.525i)9-s + (−0.150 − 0.261i)11-s + (−0.730 + 0.682i)13-s + (0.179 + 0.310i)15-s + (−0.744 + 1.28i)17-s + (0.0429 − 0.0743i)19-s + 0.590·21-s + (0.191 + 0.332i)23-s − 0.671·25-s − 1.00·27-s + (0.767 + 1.32i)29-s − 1.15·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.476 - 0.879i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.476 - 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(572\)    =    \(2^{2} \cdot 11 \cdot 13\)
Sign: $-0.476 - 0.879i$
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: \((2,\ 572,\ (\ :1/2),\ -0.476 - 0.879i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.218253 + 0.366350i\)
\(L(\frac12)\) \(\approx\) \(0.218253 + 0.366350i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.63 - 2.46i)T \)
good3 \( 1 + (0.542 + 0.939i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + 1.28T + 5T^{2} \)
7 \( 1 + (1.24 - 2.16i)T + (-3.5 - 6.06i)T^{2} \)
17 \( 1 + (3.07 - 5.31i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.187 + 0.323i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.919 - 1.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.13 - 7.15i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 6.44T + 31T^{2} \)
37 \( 1 + (-1.17 - 2.03i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.98 - 3.43i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.14 - 3.70i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 9.03T + 47T^{2} \)
53 \( 1 - 12.6T + 53T^{2} \)
59 \( 1 + (-4.07 + 7.06i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.18 + 5.51i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.93 + 10.2i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (3.27 - 5.66i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 14.8T + 73T^{2} \)
79 \( 1 + 3.17T + 79T^{2} \)
83 \( 1 + 17.3T + 83T^{2} \)
89 \( 1 + (0.905 + 1.56i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.68 + 6.37i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.30441738594989917100645823851, −10.10833635397663876385960694489, −9.191660672833116933103979717904, −8.417112465533838020640547754251, −7.29617524615461683454225628988, −6.55223521324516840998006502167, −5.70403665965159689164927581686, −4.40196301097176250629688598859, −3.26305308042808821902147044487, −1.78027023079674246927348226834, 0.24183453937500410020522831945, 2.52110597391487397835361088013, 3.93516291790184322814925325552, 4.65868131528922332143339805854, 5.66892846344280693768879887358, 7.18825230514068449296410662760, 7.45783081166506799047135534102, 8.734112185747987609026196401606, 9.989255675206709271538695990227, 10.20134471030637958046561045237

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