Properties

Label 2-1440-1.1-c3-0-21
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: $\chi_{1440} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1440,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(2.901097428\)
\(L(\frac12)\) \(\approx\) \(2.901097428\)
\(L(\frac{5}{2})\) 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

−8.983627928779647567816298626939, −8.427838331288779887210326280015, −7.54135544425348171218500778432, −6.89884173275173622235119617998, −5.64104990070064938606814840571, −4.96114164849100291571240614971, −4.35559868509634142516369702348, −2.79402996792555998509872273302, −1.97741141288338885018042808299, −0.867507139521931367981835595353, 0.867507139521931367981835595353, 1.97741141288338885018042808299, 2.79402996792555998509872273302, 4.35559868509634142516369702348, 4.96114164849100291571240614971, 5.64104990070064938606814840571, 6.89884173275173622235119617998, 7.54135544425348171218500778432, 8.427838331288779887210326280015, 8.983627928779647567816298626939

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