Properties

Label 2-570-95.18-c1-0-19
Degree $2$
Conductor $570$
Sign $-0.788 - 0.615i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.788 - 0.615i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.788 - 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.788 - 0.615i$
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: \((2,\ 570,\ (\ :1/2),\ -0.788 - 0.615i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0788471 + 0.229152i\)
\(L(\frac12)\) \(\approx\) \(0.0788471 + 0.229152i\)
\(L(\frac{3}{2})\) 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

−10.26578343823567061560179337965, −9.071180250463829314777410766388, −8.578685695952325229402552263336, −7.78508007536442461012002463332, −6.90020834977339454109445618484, −5.46062372940993664647818250631, −4.35110296535853345381372354861, −3.09678994177664536176657466006, −1.93730779021921061175212371712, −0.14336639912651409025233632331, 2.43495360141654852813626981822, 3.52453243666616173872466582710, 4.78849517007756899274998765658, 5.89618932258077967647120110086, 7.09470675179528863100960790530, 7.73726650929331890216209127313, 8.400016251607443526263220056830, 9.804486599813756921522399171141, 10.04678555425335439041388444647, 11.09102055545469856709326755637

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