Properties

Label 2-570-95.18-c1-0-0
Degree $2$
Conductor $570$
Sign $-0.699 + 0.714i$
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 + (−0.707 + 0.707i)3-s + 1.00i·4-s + (−1.29 − 1.81i)5-s − 1.00·6-s + (−0.728 + 0.728i)7-s + (−0.707 + 0.707i)8-s − 1.00i·9-s + (0.367 − 2.20i)10-s − 4.80·11-s + (−0.707 − 0.707i)12-s + (0.531 − 0.531i)13-s − 1.03·14-s + (2.20 + 0.367i)15-s − 1.00·16-s + (−3.72 + 3.72i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (−0.408 + 0.408i)3-s + 0.500i·4-s + (−0.581 − 0.813i)5-s − 0.408·6-s + (−0.275 + 0.275i)7-s + (−0.250 + 0.250i)8-s − 0.333i·9-s + (0.116 − 0.697i)10-s − 1.44·11-s + (−0.204 − 0.204i)12-s + (0.147 − 0.147i)13-s − 0.275·14-s + (0.569 + 0.0948i)15-s − 0.250·16-s + (−0.904 + 0.904i)17-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.699 + 0.714i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (493, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ -0.699 + 0.714i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00218473 - 0.00519485i\)
\(L(\frac12)\) \(\approx\) \(0.00218473 - 0.00519485i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 + (1.29 + 1.81i)T \)
19 \( 1 + (2.90 + 3.24i)T \)
good7 \( 1 + (0.728 - 0.728i)T - 7iT^{2} \)
11 \( 1 + 4.80T + 11T^{2} \)
13 \( 1 + (-0.531 + 0.531i)T - 13iT^{2} \)
17 \( 1 + (3.72 - 3.72i)T - 17iT^{2} \)
23 \( 1 + (4.07 + 4.07i)T + 23iT^{2} \)
29 \( 1 - 0.494T + 29T^{2} \)
31 \( 1 + 8.62iT - 31T^{2} \)
37 \( 1 + (-5.47 - 5.47i)T + 37iT^{2} \)
41 \( 1 - 5.82iT - 41T^{2} \)
43 \( 1 + (3.04 + 3.04i)T + 43iT^{2} \)
47 \( 1 + (-0.910 + 0.910i)T - 47iT^{2} \)
53 \( 1 + (-3.53 + 3.53i)T - 53iT^{2} \)
59 \( 1 + 12.1T + 59T^{2} \)
61 \( 1 + 5.32T + 61T^{2} \)
67 \( 1 + (-9.19 - 9.19i)T + 67iT^{2} \)
71 \( 1 - 2.06iT - 71T^{2} \)
73 \( 1 + (-3.31 - 3.31i)T + 73iT^{2} \)
79 \( 1 - 2.77T + 79T^{2} \)
83 \( 1 + (3.57 + 3.57i)T + 83iT^{2} \)
89 \( 1 - 1.08T + 89T^{2} \)
97 \( 1 + (-2.81 - 2.81i)T + 97iT^{2} \)
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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

−11.32234450347974121282304871838, −10.55360976554263578791046331210, −9.449714831877030528329640116760, −8.392422641969562982880866205074, −7.915034131188319526453561611679, −6.54402450127290885317087292650, −5.72079071577174092515363633519, −4.72276548341221147622890266208, −4.08862849583636702241095987383, −2.57846328624266421113887293600, 0.00263088369157415195831078720, 2.17131990697141187011829623520, 3.25230625540114771749793172036, 4.40377541596798577181444931057, 5.51984112973059459114140750379, 6.52747319634078099693437429276, 7.35935822109705020519112878038, 8.211374945153900454767527740096, 9.620835660875140613170696567044, 10.69926625732343971349397124406

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