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Properties

Label 2-270480-1.1-c1-0-46
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 5-s + 9-s − 2·11-s + 2·13-s + 15-s − 2·19-s + 23-s + 25-s − 27-s − 2·31-s + 2·33-s − 2·37-s − 2·39-s + 6·41-s − 2·43-s − 45-s − 2·47-s − 10·53-s + 2·55-s + 2·57-s + 4·59-s − 2·61-s − 2·65-s + 14·67-s − 69-s + 14·71-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 1/3·9-s − 0.603·11-s + 0.554·13-s + 0.258·15-s − 0.458·19-s + 0.208·23-s + 1/5·25-s − 0.192·27-s − 0.359·31-s + 0.348·33-s − 0.328·37-s − 0.320·39-s + 0.937·41-s − 0.304·43-s − 0.149·45-s − 0.291·47-s − 1.37·53-s + 0.269·55-s + 0.264·57-s + 0.520·59-s − 0.256·61-s − 0.248·65-s + 1.71·67-s − 0.120·69-s + 1.66·71-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: \(0\)
Selberg data: \((2,\ 270480,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.677232935\)
\(L(\frac12)\) \(\approx\) \(1.677232935\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 - 14 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 10 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−12.73710896667384, −12.46915755306280, −11.78063513654553, −11.30697005224982, −10.97320424280993, −10.69106381010412, −10.03849388177577, −9.605290438093745, −9.128025859921175, −8.411181783395320, −8.207543074824386, −7.605410323104839, −7.143870748827458, −6.578729688138828, −6.199625500320808, −5.637307202330326, −5.056620458487953, −4.746118904556150, −4.062760071976400, −3.567338813611739, −3.097130901894134, −2.258075309330098, −1.831988406168384, −0.9070239821559891, −0.4486147127824234, 0.4486147127824234, 0.9070239821559891, 1.831988406168384, 2.258075309330098, 3.097130901894134, 3.567338813611739, 4.062760071976400, 4.746118904556150, 5.056620458487953, 5.637307202330326, 6.199625500320808, 6.578729688138828, 7.143870748827458, 7.605410323104839, 8.207543074824386, 8.411181783395320, 9.128025859921175, 9.605290438093745, 10.03849388177577, 10.69106381010412, 10.97320424280993, 11.30697005224982, 11.78063513654553, 12.46915755306280, 12.73710896667384

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