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 + 4·7-s + 8-s + 9-s + 10-s − 4·11-s + 12-s − 2·13-s + 4·14-s + 15-s + 16-s − 2·17-s + 18-s − 19-s + 20-s + 4·21-s − 4·22-s − 8·23-s + 24-s + 25-s − 2·26-s + 27-s + 4·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 1.51·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 1.20·11-s + 0.288·12-s − 0.554·13-s + 1.06·14-s + 0.258·15-s + 1/4·16-s − 0.485·17-s + 0.235·18-s − 0.229·19-s + 0.223·20-s + 0.872·21-s − 0.852·22-s − 1.66·23-s + 0.204·24-s + 1/5·25-s − 0.392·26-s + 0.192·27-s + 0.755·28-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

\(L(1)\) \(\approx\) \(3.000997289\)
\(L(\frac12)\) \(\approx\) \(3.000997289\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−19.51027039200021, −18.48571763576448, −17.78367941408843, −17.25577148950233, −15.93088362998086, −15.48880970225319, −14.53824436158568, −14.03288514428694, −13.46207331498960, −12.42303624137739, −11.75881600526979, −10.61384993022892, −10.18247412149497, −8.758690453071398, −8.017630740750051, −7.286076146576370, −5.961721556661071, −4.985959955018035, −4.291556726395254, −2.696999524600024, −1.866242400304681, 1.866242400304681, 2.696999524600024, 4.291556726395254, 4.985959955018035, 5.961721556661071, 7.286076146576370, 8.017630740750051, 8.758690453071398, 10.18247412149497, 10.61384993022892, 11.75881600526979, 12.42303624137739, 13.46207331498960, 14.03288514428694, 14.53824436158568, 15.48880970225319, 15.93088362998086, 17.25577148950233, 17.78367941408843, 18.48571763576448, 19.51027039200021

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