Properties

Degree $2$
Conductor $1470$
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 + 2·11-s − 12-s + 2·13-s − 15-s + 16-s − 4·17-s + 18-s + 20-s + 2·22-s + 8·23-s − 24-s + 25-s + 2·26-s − 27-s − 30-s − 2·31-s + 32-s − 2·33-s − 4·34-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 + 0.603·11-s − 0.288·12-s + 0.554·13-s − 0.258·15-s + 1/4·16-s − 0.970·17-s + 0.235·18-s + 0.223·20-s + 0.426·22-s + 1.66·23-s − 0.204·24-s + 1/5·25-s + 0.392·26-s − 0.192·27-s − 0.182·30-s − 0.359·31-s + 0.176·32-s − 0.348·33-s − 0.685·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{1470} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1470,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.551067624\)
\(L(\frac12)\) \(\approx\) \(2.551067624\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
7 \( 1 \)
good11 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−19.39625080344623, −18.68035148531492, −17.93698725487591, −17.24780430819045, −16.68822746594272, −16.02424639830707, −15.14064646994985, −14.75161379831115, −13.63687570187674, −13.33758186814648, −12.53254744353934, −11.80015094139389, −10.98421581453641, −10.66494500750606, −9.438318904212610, −8.893883320761320, −7.674526691041861, −6.714480302636199, −6.268660335553761, −5.322340607290499, −4.563611785787096, −3.617844260874099, −2.418463342132777, −1.148386165095789, 1.148386165095789, 2.418463342132777, 3.617844260874099, 4.563611785787096, 5.322340607290499, 6.268660335553761, 6.714480302636199, 7.674526691041861, 8.893883320761320, 9.438318904212610, 10.66494500750606, 10.98421581453641, 11.80015094139389, 12.53254744353934, 13.33758186814648, 13.63687570187674, 14.75161379831115, 15.14064646994985, 16.02424639830707, 16.68822746594272, 17.24780430819045, 17.93698725487591, 18.68035148531492, 19.39625080344623

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