Properties

Label 2-1440-40.19-c0-0-1
Degree $2$
Conductor $1440$
Sign $1$
Analytic cond. $0.718653$
Root an. cond. $0.847734$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s + 2·19-s − 2·23-s + 25-s + 2·47-s − 49-s − 2·53-s + 2·95-s − 2·115-s + ⋯
L(s)  = 1  + 5-s + 2·19-s − 2·23-s + 25-s + 2·47-s − 49-s − 2·53-s + 2·95-s − 2·115-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1440\)    =    \(2^{5} \cdot 3^{2} \cdot 5\)
Sign: $1$
Analytic conductor: \(0.718653\)
Root analytic conductor: \(0.847734\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1440} (559, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1440,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.315142129\)
\(L(\frac12)\) \(\approx\) \(1.315142129\)
\(L(1)\) 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 \( 1 \)
5 \( 1 - T \)
good7 \( 1 + T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + T^{2} \)
17 \( ( 1 - T )( 1 + T ) \)
19 \( ( 1 - T )^{2} \)
23 \( ( 1 + T )^{2} \)
29 \( ( 1 - T )( 1 + T ) \)
31 \( ( 1 - T )( 1 + T ) \)
37 \( 1 + T^{2} \)
41 \( 1 + T^{2} \)
43 \( ( 1 - T )( 1 + T ) \)
47 \( ( 1 - T )^{2} \)
53 \( ( 1 + T )^{2} \)
59 \( 1 + T^{2} \)
61 \( ( 1 - T )( 1 + T ) \)
67 \( ( 1 - T )( 1 + T ) \)
71 \( ( 1 - T )( 1 + T ) \)
73 \( ( 1 - T )( 1 + T ) \)
79 \( ( 1 - T )( 1 + T ) \)
83 \( ( 1 - T )( 1 + T ) \)
89 \( 1 + T^{2} \)
97 \( ( 1 - T )( 1 + T ) \)
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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

−9.722412190339449603249540660149, −9.138917917560340170464046867076, −8.071837092811644793466867458103, −7.35005352982336358550583299558, −6.29749834516950681809996922649, −5.67604381279566642757589960947, −4.83932799761420115755833141631, −3.63318650069629684811236049942, −2.56091313732365169023556439842, −1.43303423446179113819549012328, 1.43303423446179113819549012328, 2.56091313732365169023556439842, 3.63318650069629684811236049942, 4.83932799761420115755833141631, 5.67604381279566642757589960947, 6.29749834516950681809996922649, 7.35005352982336358550583299558, 8.071837092811644793466867458103, 9.138917917560340170464046867076, 9.722412190339449603249540660149

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