Properties

Label 2-1470-1.1-c3-0-29
Degree $2$
Conductor $1470$
Sign $1$
Analytic cond. $86.7328$
Root an. cond. $9.31304$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 3·3-s + 4·4-s − 5·5-s + 6·6-s + 8·8-s + 9·9-s − 10·10-s − 19·11-s + 12·12-s + 33·13-s − 15·15-s + 16·16-s + 64·17-s + 18·18-s − 141·19-s − 20·20-s − 38·22-s − 51·23-s + 24·24-s + 25·25-s + 66·26-s + 27·27-s + 216·29-s − 30·30-s + 290·31-s + 32·32-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 − 0.520·11-s + 0.288·12-s + 0.704·13-s − 0.258·15-s + 1/4·16-s + 0.913·17-s + 0.235·18-s − 1.70·19-s − 0.223·20-s − 0.368·22-s − 0.462·23-s + 0.204·24-s + 1/5·25-s + 0.497·26-s + 0.192·27-s + 1.38·29-s − 0.182·30-s + 1.68·31-s + 0.176·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(86.7328\)
Root analytic conductor: \(9.31304\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1470,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(4.150580337\)
\(L(\frac12)\) \(\approx\) \(4.150580337\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - p T \)
3 \( 1 - p T \)
5 \( 1 + p T \)
7 \( 1 \)
good11 \( 1 + 19 T + p^{3} T^{2} \)
13 \( 1 - 33 T + p^{3} T^{2} \)
17 \( 1 - 64 T + p^{3} T^{2} \)
19 \( 1 + 141 T + p^{3} T^{2} \)
23 \( 1 + 51 T + p^{3} T^{2} \)
29 \( 1 - 216 T + p^{3} T^{2} \)
31 \( 1 - 290 T + p^{3} T^{2} \)
37 \( 1 + 109 T + p^{3} T^{2} \)
41 \( 1 - 457 T + p^{3} T^{2} \)
43 \( 1 - 184 T + p^{3} T^{2} \)
47 \( 1 - 313 T + p^{3} T^{2} \)
53 \( 1 + 319 T + p^{3} T^{2} \)
59 \( 1 - 44 T + p^{3} T^{2} \)
61 \( 1 + 368 T + p^{3} T^{2} \)
67 \( 1 - 216 T + p^{3} T^{2} \)
71 \( 1 + 314 T + p^{3} T^{2} \)
73 \( 1 - 602 T + p^{3} T^{2} \)
79 \( 1 - 112 T + p^{3} T^{2} \)
83 \( 1 - 712 T + p^{3} T^{2} \)
89 \( 1 + 1018 T + p^{3} T^{2} \)
97 \( 1 - 584 T + p^{3} T^{2} \)
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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

−8.950630607541310365953373570104, −8.190932487821999292969728141857, −7.66926785566122115648490713777, −6.55288556311377615273636355323, −5.92968895501135191652454313425, −4.69278796364179269200543284498, −4.08319826298312552239206874508, −3.09962742739372842869233495946, −2.26774361137910217209147479399, −0.887651083856097770918255136273, 0.887651083856097770918255136273, 2.26774361137910217209147479399, 3.09962742739372842869233495946, 4.08319826298312552239206874508, 4.69278796364179269200543284498, 5.92968895501135191652454313425, 6.55288556311377615273636355323, 7.66926785566122115648490713777, 8.190932487821999292969728141857, 8.950630607541310365953373570104

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