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Properties

Label 2-570-19.5-c1-0-7
Degree $2$
Conductor $570$
Sign $-0.838 + 0.545i$
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  + (−0.939 + 0.342i)2-s + (−0.173 + 0.984i)3-s + (0.766 − 0.642i)4-s + (−0.766 − 0.642i)5-s + (−0.173 − 0.984i)6-s + (−0.439 − 0.761i)7-s + (−0.500 + 0.866i)8-s + (−0.939 − 0.342i)9-s + (0.939 + 0.342i)10-s + (−0.266 + 0.460i)11-s + (0.5 + 0.866i)12-s + (0.141 + 0.802i)13-s + (0.673 + 0.565i)14-s + (0.766 − 0.642i)15-s + (0.173 − 0.984i)16-s + (−5.08 + 1.85i)17-s + ⋯
L(s)  = 1  + (−0.664 + 0.241i)2-s + (−0.100 + 0.568i)3-s + (0.383 − 0.321i)4-s + (−0.342 − 0.287i)5-s + (−0.0708 − 0.402i)6-s + (−0.166 − 0.287i)7-s + (−0.176 + 0.306i)8-s + (−0.313 − 0.114i)9-s + (0.297 + 0.108i)10-s + (−0.0802 + 0.138i)11-s + (0.144 + 0.249i)12-s + (0.0392 + 0.222i)13-s + (0.180 + 0.151i)14-s + (0.197 − 0.165i)15-s + (0.0434 − 0.246i)16-s + (−1.23 + 0.448i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.838 + 0.545i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.838 + 0.545i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.838 + 0.545i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ -0.838 + 0.545i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00526586 - 0.0177409i\)
\(L(\frac12)\) \(\approx\) \(0.00526586 - 0.0177409i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.939 - 0.342i)T \)
3 \( 1 + (0.173 - 0.984i)T \)
5 \( 1 + (0.766 + 0.642i)T \)
19 \( 1 + (3.37 + 2.75i)T \)
good7 \( 1 + (0.439 + 0.761i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.266 - 0.460i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.141 - 0.802i)T + (-12.2 + 4.44i)T^{2} \)
17 \( 1 + (5.08 - 1.85i)T + (13.0 - 10.9i)T^{2} \)
23 \( 1 + (4.02 - 3.37i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (5.14 + 1.87i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (1.29 + 2.24i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 2.53T + 37T^{2} \)
41 \( 1 + (-0.354 + 2.01i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (1.31 + 1.10i)T + (7.46 + 42.3i)T^{2} \)
47 \( 1 + (8.69 + 3.16i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + (2.17 - 1.82i)T + (9.20 - 52.1i)T^{2} \)
59 \( 1 + (0.152 - 0.0555i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (-1.08 + 0.907i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (10.9 + 4.00i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (5.88 + 4.93i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + (1.87 - 10.6i)T + (-68.5 - 24.9i)T^{2} \)
79 \( 1 + (-0.843 + 4.78i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (-7.02 - 12.1i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (0.699 + 3.96i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (-6.88 + 2.50i)T + (74.3 - 62.3i)T^{2} \)
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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

−10.31387159166454076721633359108, −9.419599043452763744868509716704, −8.741994572657832883763409581994, −7.83941955310342814728318582938, −6.83694257825963537865494742042, −5.90156774036923896784972100813, −4.67150883640759677114843653405, −3.75520005866464573697450137507, −2.07660420381021449072933724231, −0.01228722338216915245269650829, 1.90158095164971617548346184282, 3.03233851141583655801700967586, 4.39885244447945918953071918774, 5.93466077707259585822006244620, 6.70171692755085262373972478269, 7.65645613991352446693435074316, 8.438734480205872910128505700983, 9.236296137200767209844387931630, 10.34298027672830302129961344237, 11.06198445135653152671119238864

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