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Properties

Label 2-572-13.4-c1-0-10
Degree $2$
Conductor $572$
Sign $-0.631 + 0.775i$
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  + (1.44 − 2.50i)3-s − 0.307i·5-s + (−1.50 + 0.869i)7-s + (−2.67 − 4.63i)9-s + (−0.866 − 0.5i)11-s + (−1.23 − 3.38i)13-s + (−0.770 − 0.444i)15-s + (−2.39 − 4.14i)17-s + (−0.414 + 0.239i)19-s + 5.02i·21-s + (3.80 − 6.59i)23-s + 4.90·25-s − 6.78·27-s + (−2.94 + 5.10i)29-s − 0.819i·31-s + ⋯
L(s)  = 1  + (0.834 − 1.44i)3-s − 0.137i·5-s + (−0.569 + 0.328i)7-s + (−0.891 − 1.54i)9-s + (−0.261 − 0.150i)11-s + (−0.343 − 0.939i)13-s + (−0.198 − 0.114i)15-s + (−0.580 − 1.00i)17-s + (−0.0951 + 0.0549i)19-s + 1.09i·21-s + (0.794 − 1.37i)23-s + 0.981·25-s − 1.30·27-s + (−0.547 + 0.948i)29-s − 0.147i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.631 + 0.775i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.631 + 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(572\)    =    \(2^{2} \cdot 11 \cdot 13\)
Sign: $-0.631 + 0.775i$
Analytic conductor: \(4.56744\)
Root analytic conductor: \(2.13715\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{572} (485, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 572,\ (\ :1/2),\ -0.631 + 0.775i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.677020 - 1.42546i\)
\(L(\frac12)\) \(\approx\) \(0.677020 - 1.42546i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 + 0.5i)T \)
13 \( 1 + (1.23 + 3.38i)T \)
good3 \( 1 + (-1.44 + 2.50i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + 0.307iT - 5T^{2} \)
7 \( 1 + (1.50 - 0.869i)T + (3.5 - 6.06i)T^{2} \)
17 \( 1 + (2.39 + 4.14i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.414 - 0.239i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.80 + 6.59i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.94 - 5.10i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 0.819iT - 31T^{2} \)
37 \( 1 + (-7.49 - 4.32i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.16 - 1.82i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.20 + 5.55i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 9.33iT - 47T^{2} \)
53 \( 1 - 6.39T + 53T^{2} \)
59 \( 1 + (-4.87 + 2.81i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.27 - 9.13i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (10.4 + 6.00i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2.52 + 1.46i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 8.15iT - 73T^{2} \)
79 \( 1 - 6.15T + 79T^{2} \)
83 \( 1 + 13.8iT - 83T^{2} \)
89 \( 1 + (-5.99 - 3.45i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7.14 + 4.12i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.39083747135674394694689372467, −9.175671570554603589648058073158, −8.637247491420119147288968040003, −7.68167375212362422557832597463, −6.96197284664792258054055376616, −6.13543160590802350101661103304, −4.86451031739979422325931582987, −3.05166000693535397282418940821, −2.51561351734734872902675814381, −0.811334006406307818108047023163, 2.32265733297584490108136053327, 3.52824767259275891819763216312, 4.21701464979217332587685283724, 5.23255182018837610174661887321, 6.56232361687518866986255495833, 7.63441985033430809538319211445, 8.729142715809238050801757651911, 9.373759775668255501841822072811, 10.02171967282462122497159135992, 10.80936695189012037310434385463

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