Properties

Label 2-570-15.8-c1-0-21
Degree $2$
Conductor $570$
Sign $-0.104 - 0.994i$
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  + (0.707 + 0.707i)2-s + (1.59 + 0.668i)3-s + 1.00i·4-s + (0.595 + 2.15i)5-s + (0.656 + 1.60i)6-s + (−0.0702 + 0.0702i)7-s + (−0.707 + 0.707i)8-s + (2.10 + 2.13i)9-s + (−1.10 + 1.94i)10-s − 3.25i·11-s + (−0.668 + 1.59i)12-s + (1.84 + 1.84i)13-s − 0.0993·14-s + (−0.489 + 3.84i)15-s − 1.00·16-s + (−3.76 − 3.76i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (0.922 + 0.386i)3-s + 0.500i·4-s + (0.266 + 0.963i)5-s + (0.268 + 0.654i)6-s + (−0.0265 + 0.0265i)7-s + (−0.250 + 0.250i)8-s + (0.701 + 0.712i)9-s + (−0.348 + 0.615i)10-s − 0.981i·11-s + (−0.193 + 0.461i)12-s + (0.512 + 0.512i)13-s − 0.0265·14-s + (−0.126 + 0.991i)15-s − 0.250·16-s + (−0.913 − 0.913i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.104 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.104 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.104 - 0.994i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (533, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ -0.104 - 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.75021 + 1.94457i\)
\(L(\frac12)\) \(\approx\) \(1.75021 + 1.94457i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 0.707i)T \)
3 \( 1 + (-1.59 - 0.668i)T \)
5 \( 1 + (-0.595 - 2.15i)T \)
19 \( 1 + iT \)
good7 \( 1 + (0.0702 - 0.0702i)T - 7iT^{2} \)
11 \( 1 + 3.25iT - 11T^{2} \)
13 \( 1 + (-1.84 - 1.84i)T + 13iT^{2} \)
17 \( 1 + (3.76 + 3.76i)T + 17iT^{2} \)
23 \( 1 + (-1.02 + 1.02i)T - 23iT^{2} \)
29 \( 1 + 6.98T + 29T^{2} \)
31 \( 1 - 9.93T + 31T^{2} \)
37 \( 1 + (3.00 - 3.00i)T - 37iT^{2} \)
41 \( 1 + 8.72iT - 41T^{2} \)
43 \( 1 + (2.35 + 2.35i)T + 43iT^{2} \)
47 \( 1 + (-0.871 - 0.871i)T + 47iT^{2} \)
53 \( 1 + (-8.49 + 8.49i)T - 53iT^{2} \)
59 \( 1 - 2.24T + 59T^{2} \)
61 \( 1 - 8.33T + 61T^{2} \)
67 \( 1 + (-2.25 + 2.25i)T - 67iT^{2} \)
71 \( 1 - 6.67iT - 71T^{2} \)
73 \( 1 + (4.70 + 4.70i)T + 73iT^{2} \)
79 \( 1 + 9.81iT - 79T^{2} \)
83 \( 1 + (10.5 - 10.5i)T - 83iT^{2} \)
89 \( 1 + 0.330T + 89T^{2} \)
97 \( 1 + (3.68 - 3.68i)T - 97iT^{2} \)
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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

−11.00655200620954363190699820838, −10.03245803138131524212351214503, −9.033774484628149222703523727872, −8.382165687834170464767747764895, −7.23597566489371637481462611300, −6.59399793456274311219032229528, −5.44895320405630696835833492919, −4.20473621825487821025733182199, −3.26994785915438769120785310448, −2.33783159813236792457974507235, 1.35313503065849172419934258402, 2.36828385180400662837740707549, 3.79006875614173213337794408165, 4.58980003705701652290020380574, 5.81581271527291650673811979893, 6.87070664330110490415933806563, 8.058646404748285682198362712068, 8.757788363261234733127804811608, 9.654478749299969731845858389081, 10.33241239492787261404073978916

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