Properties

Label 2-1440-1.1-c3-0-13
Degree $2$
Conductor $1440$
Sign $1$
Analytic cond. $84.9627$
Root an. cond. $9.21752$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5·5-s − 12·7-s + 24·11-s + 38·13-s + 6·17-s + 104·19-s − 100·23-s + 25·25-s − 230·29-s − 56·31-s + 60·35-s + 190·37-s − 202·41-s − 148·43-s − 124·47-s − 199·49-s − 206·53-s − 120·55-s + 128·59-s + 190·61-s − 190·65-s − 204·67-s + 440·71-s + 1.21e3·73-s − 288·77-s + 816·79-s + 1.41e3·83-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.647·7-s + 0.657·11-s + 0.810·13-s + 0.0856·17-s + 1.25·19-s − 0.906·23-s + 1/5·25-s − 1.47·29-s − 0.324·31-s + 0.289·35-s + 0.844·37-s − 0.769·41-s − 0.524·43-s − 0.384·47-s − 0.580·49-s − 0.533·53-s − 0.294·55-s + 0.282·59-s + 0.398·61-s − 0.362·65-s − 0.371·67-s + 0.735·71-s + 1.93·73-s − 0.426·77-s + 1.16·79-s + 1.86·83-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: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1440,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.720260743\)
\(L(\frac12)\) \(\approx\) \(1.720260743\)
\(L(\frac{5}{2})\) 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 \( 1 + p T \)
good7 \( 1 + 12 T + p^{3} T^{2} \)
11 \( 1 - 24 T + p^{3} T^{2} \)
13 \( 1 - 38 T + p^{3} T^{2} \)
17 \( 1 - 6 T + p^{3} T^{2} \)
19 \( 1 - 104 T + p^{3} T^{2} \)
23 \( 1 + 100 T + p^{3} T^{2} \)
29 \( 1 + 230 T + p^{3} T^{2} \)
31 \( 1 + 56 T + p^{3} T^{2} \)
37 \( 1 - 190 T + p^{3} T^{2} \)
41 \( 1 + 202 T + p^{3} T^{2} \)
43 \( 1 + 148 T + p^{3} T^{2} \)
47 \( 1 + 124 T + p^{3} T^{2} \)
53 \( 1 + 206 T + p^{3} T^{2} \)
59 \( 1 - 128 T + p^{3} T^{2} \)
61 \( 1 - 190 T + p^{3} T^{2} \)
67 \( 1 + 204 T + p^{3} T^{2} \)
71 \( 1 - 440 T + p^{3} T^{2} \)
73 \( 1 - 1210 T + p^{3} T^{2} \)
79 \( 1 - 816 T + p^{3} T^{2} \)
83 \( 1 - 1412 T + p^{3} T^{2} \)
89 \( 1 - 214 T + p^{3} T^{2} \)
97 \( 1 - 1202 T + p^{3} T^{2} \)
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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

−9.342291157583155289271504499542, −8.297240907474522208917927019625, −7.59980455529855389380735656028, −6.67473525946051834617650169888, −5.98095234744875611285998651900, −4.99696382085790261209211444229, −3.75534862255638371370750616737, −3.37542314597662364538924813505, −1.87311514469507278264387557306, −0.65128941001997020951646049991, 0.65128941001997020951646049991, 1.87311514469507278264387557306, 3.37542314597662364538924813505, 3.75534862255638371370750616737, 4.99696382085790261209211444229, 5.98095234744875611285998651900, 6.67473525946051834617650169888, 7.59980455529855389380735656028, 8.297240907474522208917927019625, 9.342291157583155289271504499542

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