Properties

Label 2-1470-1.1-c3-0-68
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 $1$

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 + 12·11-s + 12·12-s − 2·13-s + 15·15-s + 16·16-s + 18·17-s − 18·18-s − 56·19-s + 20·20-s − 24·22-s − 156·23-s − 24·24-s + 25·25-s + 4·26-s + 27·27-s − 186·29-s − 30·30-s + 52·31-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 + 0.328·11-s + 0.288·12-s − 0.0426·13-s + 0.258·15-s + 1/4·16-s + 0.256·17-s − 0.235·18-s − 0.676·19-s + 0.223·20-s − 0.232·22-s − 1.41·23-s − 0.204·24-s + 1/5·25-s + 0.0301·26-s + 0.192·27-s − 1.19·29-s − 0.182·30-s + 0.301·31-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: \(1\)
Selberg data: \((2,\ 1470,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 - 12 T + p^{3} T^{2} \)
13 \( 1 + 2 T + p^{3} T^{2} \)
17 \( 1 - 18 T + p^{3} T^{2} \)
19 \( 1 + 56 T + p^{3} T^{2} \)
23 \( 1 + 156 T + p^{3} T^{2} \)
29 \( 1 + 186 T + p^{3} T^{2} \)
31 \( 1 - 52 T + p^{3} T^{2} \)
37 \( 1 + 178 T + p^{3} T^{2} \)
41 \( 1 - 138 T + p^{3} T^{2} \)
43 \( 1 + 412 T + p^{3} T^{2} \)
47 \( 1 - 456 T + p^{3} T^{2} \)
53 \( 1 + 198 T + p^{3} T^{2} \)
59 \( 1 + 348 T + p^{3} T^{2} \)
61 \( 1 + 110 T + p^{3} T^{2} \)
67 \( 1 + 196 T + p^{3} T^{2} \)
71 \( 1 + 936 T + p^{3} T^{2} \)
73 \( 1 + 542 T + p^{3} T^{2} \)
79 \( 1 - 992 T + p^{3} T^{2} \)
83 \( 1 - 276 T + p^{3} T^{2} \)
89 \( 1 + 630 T + p^{3} T^{2} \)
97 \( 1 + 110 T + p^{3} T^{2} \)
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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

−8.813106417143478097037545597810, −8.048032859015243871711681595642, −7.31110220765731530170985723764, −6.39959106386420823447145973759, −5.63986886928639763555082630157, −4.36087778961802358995916971978, −3.38101008602024294046413920390, −2.25324618020060436337543677924, −1.48593178852078762408526314045, 0, 1.48593178852078762408526314045, 2.25324618020060436337543677924, 3.38101008602024294046413920390, 4.36087778961802358995916971978, 5.63986886928639763555082630157, 6.39959106386420823447145973759, 7.31110220765731530170985723764, 8.048032859015243871711681595642, 8.813106417143478097037545597810

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