Properties

Label 2-1440-1.1-c1-0-9
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 + 2·7-s + 2·11-s − 2·17-s + 4·19-s + 25-s + 2·29-s + 8·31-s + 2·35-s − 4·37-s − 8·41-s + 8·43-s − 8·47-s − 3·49-s + 10·53-s + 2·55-s − 6·59-s + 2·61-s + 12·67-s + 12·71-s − 2·73-s + 4·77-s + 8·79-s − 4·83-s − 2·85-s − 12·89-s + 4·95-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.755·7-s + 0.603·11-s − 0.485·17-s + 0.917·19-s + 1/5·25-s + 0.371·29-s + 1.43·31-s + 0.338·35-s − 0.657·37-s − 1.24·41-s + 1.21·43-s − 1.16·47-s − 3/7·49-s + 1.37·53-s + 0.269·55-s − 0.781·59-s + 0.256·61-s + 1.46·67-s + 1.42·71-s − 0.234·73-s + 0.455·77-s + 0.900·79-s − 0.439·83-s − 0.216·85-s − 1.27·89-s + 0.410·95-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1440,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.116841543\)
\(L(\frac12)\) \(\approx\) \(2.116841543\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−9.567162894678014522729592248666, −8.687678118107818094251078951166, −8.044486700043922892061139958060, −7.02269464275919322402346076699, −6.31857937347981786877789666682, −5.28368610102458007670837638108, −4.59174388082872419767425141006, −3.46292942161717526500805424747, −2.25866714014751539772152464560, −1.14198203052138198817329951084, 1.14198203052138198817329951084, 2.25866714014751539772152464560, 3.46292942161717526500805424747, 4.59174388082872419767425141006, 5.28368610102458007670837638108, 6.31857937347981786877789666682, 7.02269464275919322402346076699, 8.044486700043922892061139958060, 8.687678118107818094251078951166, 9.567162894678014522729592248666

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