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Properties

Label 2-572-13.10-c1-0-5
Degree $2$
Conductor $572$
Sign $0.758 + 0.651i$
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  + (−0.780 − 1.35i)3-s − 0.661i·5-s + (3.95 + 2.28i)7-s + (0.282 − 0.489i)9-s + (0.866 − 0.5i)11-s + (−0.622 + 3.55i)13-s + (−0.893 + 0.516i)15-s + (0.341 − 0.592i)17-s + (2.99 + 1.72i)19-s − 7.11i·21-s + (−1.71 − 2.96i)23-s + 4.56·25-s − 5.56·27-s + (2.34 + 4.06i)29-s − 8.47i·31-s + ⋯
L(s)  = 1  + (−0.450 − 0.780i)3-s − 0.295i·5-s + (1.49 + 0.862i)7-s + (0.0941 − 0.163i)9-s + (0.261 − 0.150i)11-s + (−0.172 + 0.985i)13-s + (−0.230 + 0.133i)15-s + (0.0829 − 0.143i)17-s + (0.686 + 0.396i)19-s − 1.55i·21-s + (−0.356 − 0.618i)23-s + 0.912·25-s − 1.07·27-s + (0.435 + 0.754i)29-s − 1.52i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.758 + 0.651i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.758 + 0.651i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(572\)    =    \(2^{2} \cdot 11 \cdot 13\)
Sign: $0.758 + 0.651i$
Analytic conductor: \(4.56744\)
Root analytic conductor: \(2.13715\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{572} (309, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 572,\ (\ :1/2),\ 0.758 + 0.651i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.43488 - 0.531788i\)
\(L(\frac12)\) \(\approx\) \(1.43488 - 0.531788i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.622 - 3.55i)T \)
good3 \( 1 + (0.780 + 1.35i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + 0.661iT - 5T^{2} \)
7 \( 1 + (-3.95 - 2.28i)T + (3.5 + 6.06i)T^{2} \)
17 \( 1 + (-0.341 + 0.592i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.99 - 1.72i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.71 + 2.96i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.34 - 4.06i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 8.47iT - 31T^{2} \)
37 \( 1 + (-7.15 + 4.12i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.354 - 0.204i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.98 + 3.43i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 12.6iT - 47T^{2} \)
53 \( 1 + 6.70T + 53T^{2} \)
59 \( 1 + (10.6 + 6.14i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.68 - 4.65i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (11.4 - 6.59i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-12.6 - 7.27i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 12.4iT - 73T^{2} \)
79 \( 1 + 5.02T + 79T^{2} \)
83 \( 1 - 14.7iT - 83T^{2} \)
89 \( 1 + (-5.65 + 3.26i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.48 - 1.43i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.98081953222101680667289786378, −9.595970798399246167461419693049, −8.789817223975651531159830264397, −7.949729063718043547034087070666, −7.05012136940021116502435260413, −6.04471282029571436682062397976, −5.16634212770750470764349591236, −4.15646260887129207952967429552, −2.27727300778332839957839012675, −1.21153336070990248983462129848, 1.35570475778366316010137802172, 3.16523803911778724801286555810, 4.59377626760816752784727228874, 4.87724702466204585617337447904, 6.14654042905329527931075813743, 7.57127705790940128169545850187, 7.88178574782233145098079786429, 9.260524425826112226366058638525, 10.25063712903211779869477309697, 10.79209623493108563605882643078

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