Properties

Label 2-570-285.269-c1-0-31
Degree $2$
Conductor $570$
Sign $0.714 + 0.699i$
Analytic cond. $4.55147$
Root an. cond. $2.13341$
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  + (−0.642 + 0.766i)2-s + (1.61 − 0.638i)3-s + (−0.173 − 0.984i)4-s + (1.09 − 1.95i)5-s + (−0.546 + 1.64i)6-s + (1.53 − 0.888i)7-s + (0.866 + 0.500i)8-s + (2.18 − 2.05i)9-s + (0.791 + 2.09i)10-s + (−3.70 − 2.14i)11-s + (−0.908 − 1.47i)12-s + (2.04 + 0.743i)13-s + (−0.308 + 1.75i)14-s + (0.514 − 3.83i)15-s + (−0.939 + 0.342i)16-s + (−0.583 − 0.489i)17-s + ⋯
L(s)  = 1  + (−0.454 + 0.541i)2-s + (0.929 − 0.368i)3-s + (−0.0868 − 0.492i)4-s + (0.488 − 0.872i)5-s + (−0.222 + 0.671i)6-s + (0.581 − 0.335i)7-s + (0.306 + 0.176i)8-s + (0.728 − 0.685i)9-s + (0.250 + 0.661i)10-s + (−1.11 − 0.645i)11-s + (−0.262 − 0.425i)12-s + (0.566 + 0.206i)13-s + (−0.0824 + 0.467i)14-s + (0.132 − 0.991i)15-s + (−0.234 + 0.0855i)16-s + (−0.141 − 0.118i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.714 + 0.699i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.714 + 0.699i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.714 + 0.699i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 0.714 + 0.699i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.57827 - 0.644109i\)
\(L(\frac12)\) \(\approx\) \(1.57827 - 0.644109i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.642 - 0.766i)T \)
3 \( 1 + (-1.61 + 0.638i)T \)
5 \( 1 + (-1.09 + 1.95i)T \)
19 \( 1 + (4.17 + 1.26i)T \)
good7 \( 1 + (-1.53 + 0.888i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (3.70 + 2.14i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.04 - 0.743i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (0.583 + 0.489i)T + (2.95 + 16.7i)T^{2} \)
23 \( 1 + (-1.10 - 6.28i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (-1.06 + 0.894i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (4.34 - 2.50i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 7.40T + 37T^{2} \)
41 \( 1 + (-3.10 + 1.13i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (-0.00915 - 0.00161i)T + (40.4 + 14.7i)T^{2} \)
47 \( 1 + (-3.61 + 3.03i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 + (-0.204 + 0.0360i)T + (49.8 - 18.1i)T^{2} \)
59 \( 1 + (-9.23 - 7.74i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-1.87 - 10.6i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (-3.11 + 2.61i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (-2.40 + 13.6i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (-2.88 - 7.91i)T + (-55.9 + 46.9i)T^{2} \)
79 \( 1 + (-1.48 - 4.07i)T + (-60.5 + 50.7i)T^{2} \)
83 \( 1 + (-5.46 - 9.46i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-4.59 - 1.67i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (2.62 + 2.20i)T + (16.8 + 95.5i)T^{2} \)
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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

−10.42458788240157456169708153811, −9.416071486786365512721566877315, −8.725385670276863346140930767166, −8.095977534778631161939099678076, −7.34879806017040543953637721796, −6.14237055800885408991183159807, −5.17003929856313793941520054728, −4.01134612039983924167963067799, −2.36490446701742030241754071108, −1.09397215213108229211984636618, 2.05318287754833519240498608088, 2.64199562196464181351459730107, 3.92011589457997167807281635921, 5.06693536038693487883148161659, 6.48037917170147806653173432159, 7.65647909106506252066261403518, 8.265886104531112597508371033430, 9.170829380475770159478112290327, 10.10444362124022206988649203935, 10.62783089061669474094402033937

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