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 + 8-s + 9-s − 10-s + 4·11-s − 12-s + 2·13-s + 15-s + 16-s + 2·17-s + 18-s − 19-s − 20-s + 4·22-s + 4·23-s − 24-s + 25-s + 2·26-s − 27-s + 6·29-s + 30-s + 4·31-s + 32-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 1.20·11-s − 0.288·12-s + 0.554·13-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 + 0.852·22-s + 0.834·23-s − 0.204·24-s + 1/5·25-s + 0.392·26-s − 0.192·27-s + 1.11·29-s + 0.182·30-s + 0.718·31-s + 0.176·32-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

\(L(1)\) \(\approx\) \(1.912920243\)
\(L(\frac12)\) \(\approx\) \(1.912920243\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
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

−19.95527257077485, −19.44436370002367, −18.67365926662629, −17.64446662139139, −16.98549314833278, −16.22548245708322, −15.58637180097926, −14.69945914640243, −14.06736410703248, −13.09681024464080, −12.30667603466917, −11.66747212041486, −11.02323073036755, −10.08161684730657, −8.971765324735811, −7.944857545879670, −6.787881226915262, −6.255399949044747, −5.057445710796400, −4.171922998822869, −3.156560868081350, −1.296827798684915, 1.296827798684915, 3.156560868081350, 4.171922998822869, 5.057445710796400, 6.255399949044747, 6.787881226915262, 7.944857545879670, 8.971765324735811, 10.08161684730657, 11.02323073036755, 11.66747212041486, 12.30667603466917, 13.09681024464080, 14.06736410703248, 14.69945914640243, 15.58637180097926, 16.22548245708322, 16.98549314833278, 17.64446662139139, 18.67365926662629, 19.44436370002367, 19.95527257077485

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