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Properties

Label 2-572-13.3-c1-0-5
Degree $2$
Conductor $572$
Sign $0.240 + 0.970i$
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  + (0.334 + 0.579i)3-s − 4.07·5-s + (−0.950 + 1.64i)7-s + (1.27 − 2.21i)9-s + (−0.5 − 0.866i)11-s + (3.16 − 1.72i)13-s + (−1.36 − 2.35i)15-s + (1.91 − 3.31i)17-s + (2.09 − 3.62i)19-s − 1.27·21-s + (−2.68 − 4.64i)23-s + 11.5·25-s + 3.71·27-s + (−3.05 − 5.29i)29-s − 7.32·31-s + ⋯
L(s)  = 1  + (0.193 + 0.334i)3-s − 1.82·5-s + (−0.359 + 0.622i)7-s + (0.425 − 0.736i)9-s + (−0.150 − 0.261i)11-s + (0.878 − 0.477i)13-s + (−0.351 − 0.609i)15-s + (0.464 − 0.804i)17-s + (0.480 − 0.832i)19-s − 0.277·21-s + (−0.559 − 0.968i)23-s + 2.31·25-s + 0.714·27-s + (−0.568 − 0.984i)29-s − 1.31·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.240 + 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.240 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(572\)    =    \(2^{2} \cdot 11 \cdot 13\)
Sign: $0.240 + 0.970i$
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: \((2,\ 572,\ (\ :1/2),\ 0.240 + 0.970i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.658507 - 0.515264i\)
\(L(\frac12)\) \(\approx\) \(0.658507 - 0.515264i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.16 + 1.72i)T \)
good3 \( 1 + (-0.334 - 0.579i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + 4.07T + 5T^{2} \)
7 \( 1 + (0.950 - 1.64i)T + (-3.5 - 6.06i)T^{2} \)
17 \( 1 + (-1.91 + 3.31i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.09 + 3.62i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.68 + 4.64i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.05 + 5.29i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 7.32T + 31T^{2} \)
37 \( 1 + (0.945 + 1.63i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.24 + 2.15i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.53 - 6.12i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 7.24T + 47T^{2} \)
53 \( 1 + 8.63T + 53T^{2} \)
59 \( 1 + (4.87 - 8.43i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.70 + 6.41i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.78 + 4.81i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2.31 + 4.01i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 1.84T + 73T^{2} \)
79 \( 1 - 9.03T + 79T^{2} \)
83 \( 1 - 5.28T + 83T^{2} \)
89 \( 1 + (0.677 + 1.17i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.18 + 10.7i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.73634029560546051823242699261, −9.489310248990748669646378912480, −8.794031179444276396922703287920, −7.916301007530669203040466053507, −7.13112147427047745924823752783, −5.99846710969567267704357965233, −4.67994113534913815533661839336, −3.71141297510874971235231037253, −3.01585956562856012131690471113, −0.50210304871154352869441912294, 1.50704222243567259668273048306, 3.57820501046134175550194327001, 3.90031477033024196622887092630, 5.25114799388860508510538089722, 6.73638797082827524878545231667, 7.58016361164252462119878100908, 7.938255259764440732171051682653, 8.960084881348286625721021209899, 10.28689610386595549886184007441, 10.92031959811652588529462586992

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