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Properties

Label 2-270480-1.1-c1-0-235
Degree $2$
Conductor $270480$
Sign $1$
Analytic cond. $2159.79$
Root an. cond. $46.4735$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 5-s + 9-s − 11-s − 6·13-s + 15-s + 4·17-s − 19-s − 23-s + 25-s − 27-s + 33-s − 8·37-s + 6·39-s − 3·41-s − 4·43-s − 45-s − 7·47-s − 4·51-s + 11·53-s + 55-s + 57-s − 11·59-s − 5·61-s + 6·65-s − 6·67-s + 69-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 1/3·9-s − 0.301·11-s − 1.66·13-s + 0.258·15-s + 0.970·17-s − 0.229·19-s − 0.208·23-s + 1/5·25-s − 0.192·27-s + 0.174·33-s − 1.31·37-s + 0.960·39-s − 0.468·41-s − 0.609·43-s − 0.149·45-s − 1.02·47-s − 0.560·51-s + 1.51·53-s + 0.134·55-s + 0.132·57-s − 1.43·59-s − 0.640·61-s + 0.744·65-s − 0.733·67-s + 0.120·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 270480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(270480\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7^{2} \cdot 23\)
Sign: $1$
Analytic conductor: \(2159.79\)
Root analytic conductor: \(46.4735\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((2,\ 270480,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
5 \( 1 + T \)
7 \( 1 \)
23 \( 1 + T \)
good11 \( 1 + T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 7 T + p T^{2} \)
53 \( 1 - 11 T + p T^{2} \)
59 \( 1 + 11 T + p T^{2} \)
61 \( 1 + 5 T + p T^{2} \)
67 \( 1 + 6 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 - 4 T + p T^{2} \)
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

−13.24294219544869, −12.58775418664246, −12.14866682231774, −11.97906776702407, −11.61787175184484, −10.82288921614540, −10.48636966656150, −10.12988411424774, −9.606676743562674, −9.214027185608135, −8.445687360672214, −8.124039530264138, −7.446149625602080, −7.265565294569112, −6.723039766772758, −6.129440314109721, −5.513221981978193, −5.154572422588312, −4.670029084316372, −4.232157777989378, −3.424170150228236, −3.091596240249809, −2.347740165099620, −1.749526683053078, −1.068348929251433, 0, 0, 1.068348929251433, 1.749526683053078, 2.347740165099620, 3.091596240249809, 3.424170150228236, 4.232157777989378, 4.670029084316372, 5.154572422588312, 5.513221981978193, 6.129440314109721, 6.723039766772758, 7.265565294569112, 7.446149625602080, 8.124039530264138, 8.445687360672214, 9.214027185608135, 9.606676743562674, 10.12988411424774, 10.48636966656150, 10.82288921614540, 11.61787175184484, 11.97906776702407, 12.14866682231774, 12.58775418664246, 13.24294219544869

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