Properties

Label 2-1440-1.1-c1-0-8
Degree $2$
Conductor $1440$
Sign $1$
Analytic cond. $11.4984$
Root an. cond. $3.39093$
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  + 5-s + 4·11-s + 2·13-s + 2·17-s − 8·19-s + 4·23-s + 25-s + 6·29-s + 2·37-s + 6·41-s − 4·43-s − 12·47-s − 7·49-s + 6·53-s + 4·55-s + 12·59-s + 14·61-s + 2·65-s + 12·67-s + 2·73-s + 8·79-s − 4·83-s + 2·85-s − 2·89-s − 8·95-s − 14·97-s + 14·101-s + ⋯
L(s)  = 1  + 0.447·5-s + 1.20·11-s + 0.554·13-s + 0.485·17-s − 1.83·19-s + 0.834·23-s + 1/5·25-s + 1.11·29-s + 0.328·37-s + 0.937·41-s − 0.609·43-s − 1.75·47-s − 49-s + 0.824·53-s + 0.539·55-s + 1.56·59-s + 1.79·61-s + 0.248·65-s + 1.46·67-s + 0.234·73-s + 0.900·79-s − 0.439·83-s + 0.216·85-s − 0.211·89-s − 0.820·95-s − 1.42·97-s + 1.39·101-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1440\)    =    \(2^{5} \cdot 3^{2} \cdot 5\)
Sign: $1$
Analytic conductor: \(11.4984\)
Root analytic conductor: \(3.39093\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1440} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1440,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.013504913\)
\(L(\frac12)\) \(\approx\) \(2.013504913\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 - T \)
good7 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−9.547438999502026658807482308365, −8.667981626902712297406058340761, −8.176989849593365125562259771659, −6.68758445534728746261165546175, −6.54644616692428564421950720399, −5.41791684827641272932549263816, −4.40013024639620411659480510891, −3.53852999845818475814528031258, −2.28371327004468584640295662946, −1.09896676042479434195845174014, 1.09896676042479434195845174014, 2.28371327004468584640295662946, 3.53852999845818475814528031258, 4.40013024639620411659480510891, 5.41791684827641272932549263816, 6.54644616692428564421950720399, 6.68758445534728746261165546175, 8.176989849593365125562259771659, 8.667981626902712297406058340761, 9.547438999502026658807482308365

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