Properties

Label 2-1440-1.1-c3-0-8
Degree $2$
Conductor $1440$
Sign $1$
Analytic cond. $84.9627$
Root an. cond. $9.21752$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5·5-s − 31.3·7-s + 8.94·11-s − 62·13-s + 46·17-s + 107.·19-s − 192.·23-s + 25·25-s + 90·29-s − 152.·31-s − 156.·35-s − 214·37-s + 10·41-s − 67.0·43-s − 398.·47-s + 637.·49-s + 678·53-s + 44.7·55-s + 411.·59-s + 250·61-s − 310·65-s + 49.1·67-s + 366.·71-s + 522·73-s − 280·77-s + 876.·79-s − 380.·83-s + ⋯
L(s)  = 1  + 0.447·5-s − 1.69·7-s + 0.245·11-s − 1.32·13-s + 0.656·17-s + 1.29·19-s − 1.74·23-s + 0.200·25-s + 0.576·29-s − 0.880·31-s − 0.755·35-s − 0.950·37-s + 0.0380·41-s − 0.237·43-s − 1.23·47-s + 1.85·49-s + 1.75·53-s + 0.109·55-s + 0.907·59-s + 0.524·61-s − 0.591·65-s + 0.0897·67-s + 0.612·71-s + 0.836·73-s − 0.414·77-s + 1.24·79-s − 0.502·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1440\)    =    \(2^{5} \cdot 3^{2} \cdot 5\)
Sign: $1$
Analytic conductor: \(84.9627\)
Root analytic conductor: \(9.21752\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1440,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.325151525\)
\(L(\frac12)\) \(\approx\) \(1.325151525\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 - 5T \)
good7 \( 1 + 31.3T + 343T^{2} \)
11 \( 1 - 8.94T + 1.33e3T^{2} \)
13 \( 1 + 62T + 2.19e3T^{2} \)
17 \( 1 - 46T + 4.91e3T^{2} \)
19 \( 1 - 107.T + 6.85e3T^{2} \)
23 \( 1 + 192.T + 1.21e4T^{2} \)
29 \( 1 - 90T + 2.43e4T^{2} \)
31 \( 1 + 152.T + 2.97e4T^{2} \)
37 \( 1 + 214T + 5.06e4T^{2} \)
41 \( 1 - 10T + 6.89e4T^{2} \)
43 \( 1 + 67.0T + 7.95e4T^{2} \)
47 \( 1 + 398.T + 1.03e5T^{2} \)
53 \( 1 - 678T + 1.48e5T^{2} \)
59 \( 1 - 411.T + 2.05e5T^{2} \)
61 \( 1 - 250T + 2.26e5T^{2} \)
67 \( 1 - 49.1T + 3.00e5T^{2} \)
71 \( 1 - 366.T + 3.57e5T^{2} \)
73 \( 1 - 522T + 3.89e5T^{2} \)
79 \( 1 - 876.T + 4.93e5T^{2} \)
83 \( 1 + 380.T + 5.71e5T^{2} \)
89 \( 1 + 970T + 7.04e5T^{2} \)
97 \( 1 + 934T + 9.12e5T^{2} \)
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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

−9.558022948169763498860957028814, −8.417994985471116099384001005169, −7.34620666484450842011991198076, −6.77719653018365708186155486420, −5.85726824639262399200624828530, −5.18700649392820428843871395614, −3.83630600896124833998605354063, −3.08895946567299908889273426666, −2.07325714783754065014939677866, −0.54307365479155893514727022506, 0.54307365479155893514727022506, 2.07325714783754065014939677866, 3.08895946567299908889273426666, 3.83630600896124833998605354063, 5.18700649392820428843871395614, 5.85726824639262399200624828530, 6.77719653018365708186155486420, 7.34620666484450842011991198076, 8.417994985471116099384001005169, 9.558022948169763498860957028814

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