Properties

Degree $2$
Conductor $510$
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 + 17-s + 18-s + 4·19-s + 20-s + 4·22-s − 24-s + 25-s − 2·26-s − 27-s − 2·29-s − 30-s + 8·31-s + 32-s − 4·33-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 + 0.242·17-s + 0.235·18-s + 0.917·19-s + 0.223·20-s + 0.852·22-s − 0.204·24-s + 1/5·25-s − 0.392·26-s − 0.192·27-s − 0.371·29-s − 0.182·30-s + 1.43·31-s + 0.176·32-s − 0.696·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 510 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 510 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(510\)    =    \(2 \cdot 3 \cdot 5 \cdot 17\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{510} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 510,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.080987183\)
\(L(\frac12)\) \(\approx\) \(2.080987183\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
17 \( 1 - T \)
good7 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−19.66625320773818, −18.85799344200304, −17.90935553086823, −17.13374948878175, −16.64533385891158, −15.73626961964836, −14.85556121399990, −14.15127652101459, −13.43945477716384, −12.47671952501580, −11.81798364652719, −11.18171246385487, −10.00863408081627, −9.429683456004767, −8.039666398068161, −6.906046585966420, −6.231008525570851, −5.246368422485726, −4.337392339520562, −3.058091777133188, −1.453908807556173, 1.453908807556173, 3.058091777133188, 4.337392339520562, 5.246368422485726, 6.231008525570851, 6.906046585966420, 8.039666398068161, 9.429683456004767, 10.00863408081627, 11.18171246385487, 11.81798364652719, 12.47671952501580, 13.43945477716384, 14.15127652101459, 14.85556121399990, 15.73626961964836, 16.64533385891158, 17.13374948878175, 17.90935553086823, 18.85799344200304, 19.66625320773818

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