Properties

Label 2-1440-1.1-c1-0-18
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 $1$

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: $\chi_{1440} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1440,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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

−9.008791119141243942717159667737, −8.564085609686467345322463540902, −7.37242345424133052115191128471, −6.68289289853968084678108249114, −5.85349627514271365714593881393, −5.01909859274436078455320866302, −3.91480640522384398014282785525, −2.88838332187413604660369820241, −1.84983133398470330100799616342, 0, 1.84983133398470330100799616342, 2.88838332187413604660369820241, 3.91480640522384398014282785525, 5.01909859274436078455320866302, 5.85349627514271365714593881393, 6.68289289853968084678108249114, 7.37242345424133052115191128471, 8.564085609686467345322463540902, 9.008791119141243942717159667737

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