Properties

Label 2-570-19.9-c1-0-11
Degree $2$
Conductor $570$
Sign $-0.980 + 0.196i$
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  + (−0.173 − 0.984i)2-s + (0.766 − 0.642i)3-s + (−0.939 + 0.342i)4-s + (−0.939 − 0.342i)5-s + (−0.766 − 0.642i)6-s + (−0.326 + 0.565i)7-s + (0.5 + 0.866i)8-s + (0.173 − 0.984i)9-s + (−0.173 + 0.984i)10-s + (−1.43 − 2.49i)11-s + (−0.499 + 0.866i)12-s + (−1.26 − 1.06i)13-s + (0.613 + 0.223i)14-s + (−0.939 + 0.342i)15-s + (0.766 − 0.642i)16-s + (−1.26 − 7.18i)17-s + ⋯
L(s)  = 1  + (−0.122 − 0.696i)2-s + (0.442 − 0.371i)3-s + (−0.469 + 0.171i)4-s + (−0.420 − 0.152i)5-s + (−0.312 − 0.262i)6-s + (−0.123 + 0.213i)7-s + (0.176 + 0.306i)8-s + (0.0578 − 0.328i)9-s + (−0.0549 + 0.311i)10-s + (−0.434 − 0.751i)11-s + (−0.144 + 0.249i)12-s + (−0.351 − 0.294i)13-s + (0.163 + 0.0596i)14-s + (−0.242 + 0.0883i)15-s + (0.191 − 0.160i)16-s + (−0.307 − 1.74i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.980 + 0.196i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.980 + 0.196i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.980 + 0.196i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (541, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ -0.980 + 0.196i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0900416 - 0.907921i\)
\(L(\frac12)\) \(\approx\) \(0.0900416 - 0.907921i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.173 + 0.984i)T \)
3 \( 1 + (-0.766 + 0.642i)T \)
5 \( 1 + (0.939 + 0.342i)T \)
19 \( 1 + (3.79 + 2.15i)T \)
good7 \( 1 + (0.326 - 0.565i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.43 + 2.49i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.26 + 1.06i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (1.26 + 7.18i)T + (-15.9 + 5.81i)T^{2} \)
23 \( 1 + (1.61 - 0.587i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (-0.0282 + 0.160i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (0.471 - 0.817i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 7.24T + 37T^{2} \)
41 \( 1 + (0.309 - 0.259i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (-10.6 - 3.88i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (-1.00 + 5.71i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (-9.93 + 3.61i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (0.790 + 4.48i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (10.8 - 3.93i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (0.396 - 2.24i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (-4.01 - 1.45i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (-8.73 + 7.32i)T + (12.6 - 71.8i)T^{2} \)
79 \( 1 + (3.73 - 3.13i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (-2.14 + 3.71i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-1.91 - 1.60i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (2.43 + 13.8i)T + (-91.1 + 33.1i)T^{2} \)
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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

−10.42376007026106353171240187476, −9.313152913206409961394689287714, −8.727910629912058946147212059695, −7.80367573842345366745589209363, −6.93975622202683208307852400709, −5.55895120630565437815067054357, −4.47390220807510607011636566771, −3.20653929455301551457809703071, −2.35082057719640196427356398328, −0.49505754036901851570277907216, 2.11261080324893219195223820529, 3.80231476069122600728715484167, 4.44795998934861232981555917942, 5.73416648119070447452539245585, 6.78012022532417434690554365842, 7.66595791552742188101942809197, 8.407218127620115830470889600893, 9.233598599858421587123636512218, 10.35485305729759500773815961815, 10.67416349535965730606350945119

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