Properties

Label 2-570-95.49-c1-0-0
Degree $2$
Conductor $570$
Sign $-0.640 - 0.767i$
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  + (−0.866 − 0.5i)2-s + (−0.866 − 0.5i)3-s + (0.499 + 0.866i)4-s + (−0.837 + 2.07i)5-s + (0.499 + 0.866i)6-s − 0.785i·7-s − 0.999i·8-s + (0.499 + 0.866i)9-s + (1.76 − 1.37i)10-s − 0.377·11-s − 0.999i·12-s + (2.51 − 1.45i)13-s + (−0.392 + 0.680i)14-s + (1.76 − 1.37i)15-s + (−0.5 + 0.866i)16-s + (−2.45 − 1.41i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (−0.499 − 0.288i)3-s + (0.249 + 0.433i)4-s + (−0.374 + 0.927i)5-s + (0.204 + 0.353i)6-s − 0.296i·7-s − 0.353i·8-s + (0.166 + 0.288i)9-s + (0.557 − 0.435i)10-s − 0.113·11-s − 0.288i·12-s + (0.697 − 0.402i)13-s + (−0.104 + 0.181i)14-s + (0.454 − 0.355i)15-s + (−0.125 + 0.216i)16-s + (−0.595 − 0.343i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.640 - 0.767i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.640 - 0.767i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.640 - 0.767i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ -0.640 - 0.767i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.113283 + 0.242138i\)
\(L(\frac12)\) \(\approx\) \(0.113283 + 0.242138i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 + (0.866 + 0.5i)T \)
5 \( 1 + (0.837 - 2.07i)T \)
19 \( 1 + (2.82 - 3.32i)T \)
good7 \( 1 + 0.785iT - 7T^{2} \)
11 \( 1 + 0.377T + 11T^{2} \)
13 \( 1 + (-2.51 + 1.45i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.45 + 1.41i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (7.86 - 4.54i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.01 - 3.48i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 4.42T + 31T^{2} \)
37 \( 1 - 0.0967iT - 37T^{2} \)
41 \( 1 + (1.43 - 2.48i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.371 - 0.214i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (10.4 - 6.04i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.00 - 3.46i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.88 - 8.45i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.27 + 3.94i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.31 - 4.22i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-5.86 + 10.1i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-1.95 - 1.13i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.785 - 1.36i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 2.75iT - 83T^{2} \)
89 \( 1 + (-5.96 - 10.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.98 + 1.72i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.89042348659970415740135582784, −10.45756449235743937904518487520, −9.487666609621544048815017679375, −8.211234895420415552990675938813, −7.61702674542588835828023720268, −6.63464825897014033755328069819, −5.85568200115771057093357473372, −4.22329899273273051940871231321, −3.18784699242236344491500367619, −1.74774865446199518683536122956, 0.19167503216436158948729416464, 1.94415917968314343127077539860, 3.96545908697617352225225800763, 4.84807166883054307849058768871, 5.94801154189764276562417803145, 6.69774634458703548557716761506, 8.031575390144886085940919775018, 8.640147602556204189188073631013, 9.394469693780464174155759353493, 10.38416091593359195852880671897

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