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Properties

Label 2-572-13.10-c1-0-3
Degree $2$
Conductor $572$
Sign $-0.168 - 0.985i$
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  + (0.196 + 0.340i)3-s + 4.15i·5-s + (2.01 + 1.16i)7-s + (1.42 − 2.46i)9-s + (−0.866 + 0.5i)11-s + (−2.74 + 2.33i)13-s + (−1.41 + 0.816i)15-s + (0.0827 − 0.143i)17-s + (1.59 + 0.918i)19-s + 0.915i·21-s + (−0.418 − 0.724i)23-s − 12.2·25-s + 2.29·27-s + (0.469 + 0.812i)29-s + 1.37i·31-s + ⋯
L(s)  = 1  + (0.113 + 0.196i)3-s + 1.85i·5-s + (0.761 + 0.439i)7-s + (0.474 − 0.821i)9-s + (−0.261 + 0.150i)11-s + (−0.761 + 0.648i)13-s + (−0.365 + 0.210i)15-s + (0.0200 − 0.0347i)17-s + (0.364 + 0.210i)19-s + 0.199i·21-s + (−0.0871 − 0.151i)23-s − 2.44·25-s + 0.442·27-s + (0.0871 + 0.150i)29-s + 0.247i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.168 - 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.168 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(572\)    =    \(2^{2} \cdot 11 \cdot 13\)
Sign: $-0.168 - 0.985i$
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: \((2,\ 572,\ (\ :1/2),\ -0.168 - 0.985i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.985094 + 1.16769i\)
\(L(\frac12)\) \(\approx\) \(0.985094 + 1.16769i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.74 - 2.33i)T \)
good3 \( 1 + (-0.196 - 0.340i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 - 4.15iT - 5T^{2} \)
7 \( 1 + (-2.01 - 1.16i)T + (3.5 + 6.06i)T^{2} \)
17 \( 1 + (-0.0827 + 0.143i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.59 - 0.918i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.418 + 0.724i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.469 - 0.812i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 1.37iT - 31T^{2} \)
37 \( 1 + (0.644 - 0.371i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (7.53 - 4.34i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.41 + 7.65i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 9.78iT - 47T^{2} \)
53 \( 1 - 13.1T + 53T^{2} \)
59 \( 1 + (-6.81 - 3.93i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.13 - 1.96i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-10.7 + 6.23i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-7.71 - 4.45i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 14.7iT - 73T^{2} \)
79 \( 1 + 1.93T + 79T^{2} \)
83 \( 1 - 12.5iT - 83T^{2} \)
89 \( 1 + (2.14 - 1.23i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.09 + 1.78i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.92713035439576008941857557372, −10.11980851470263405537724832830, −9.469034922562036377232061063449, −8.238812744005299976320832786770, −7.17257751459439702716035106176, −6.71754793252187486412198872639, −5.52796631709214336698066930760, −4.21424806966576024476312697638, −3.12689550752253362607297265973, −2.08162228400667326799824637497, 0.898058442045763323476432998988, 2.12773609610824625719671805936, 4.06469962945353719361481000704, 5.01581093561458131543279217120, 5.41252454986375761736870478910, 7.19467502381933453233211173669, 8.022008788392977936920559542847, 8.483221649597666888589611542121, 9.608513264940876166868319616536, 10.37405154694543409712552322511

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