Properties

Degree $2$
Conductor $570$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s + 5-s − 6-s − 2·7-s + 8-s + 9-s + 10-s − 12-s + 6·13-s − 2·14-s − 15-s + 16-s + 8·17-s + 18-s + 19-s + 20-s + 2·21-s − 4·23-s − 24-s + 25-s + 6·26-s − 27-s − 2·28-s + 2·29-s − 30-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.755·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.288·12-s + 1.66·13-s − 0.534·14-s − 0.258·15-s + 1/4·16-s + 1.94·17-s + 0.235·18-s + 0.229·19-s + 0.223·20-s + 0.436·21-s − 0.834·23-s − 0.204·24-s + 1/5·25-s + 1.17·26-s − 0.192·27-s − 0.377·28-s + 0.371·29-s − 0.182·30-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{570} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.041836350\)
\(L(\frac12)\) \(\approx\) \(2.041836350\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + T \)
5 \( 1 - T \)
19 \( 1 - T \)
good7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 8 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−19.19259083843502, −18.58171186667604, −17.86907718139636, −16.76310742093829, −16.33095356278500, −15.67731078650236, −14.71763177550742, −13.70148684413015, −13.39056391131021, −12.25476197380245, −11.90080909481762, −10.63238364184484, −10.20196620364905, −9.101594919437242, −7.916353674876268, −6.811404574247936, −5.927151442085394, −5.484817272958238, −3.997866692049154, −3.136289925817132, −1.358886654275707, 1.358886654275707, 3.136289925817132, 3.997866692049154, 5.484817272958238, 5.927151442085394, 6.811404574247936, 7.916353674876268, 9.101594919437242, 10.20196620364905, 10.63238364184484, 11.90080909481762, 12.25476197380245, 13.39056391131021, 13.70148684413015, 14.71763177550742, 15.67731078650236, 16.33095356278500, 16.76310742093829, 17.86907718139636, 18.58171186667604, 19.19259083843502

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