Properties

Degree $2$
Conductor $960$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 5-s + 4·7-s + 9-s − 2·13-s + 15-s + 6·17-s − 4·19-s + 4·21-s + 25-s + 27-s + 6·29-s − 8·31-s + 4·35-s − 2·37-s − 2·39-s − 6·41-s − 4·43-s + 45-s + 9·49-s + 6·51-s + 6·53-s − 4·57-s + 10·61-s + 4·63-s − 2·65-s − 4·67-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 1.51·7-s + 1/3·9-s − 0.554·13-s + 0.258·15-s + 1.45·17-s − 0.917·19-s + 0.872·21-s + 1/5·25-s + 0.192·27-s + 1.11·29-s − 1.43·31-s + 0.676·35-s − 0.328·37-s − 0.320·39-s − 0.937·41-s − 0.609·43-s + 0.149·45-s + 9/7·49-s + 0.840·51-s + 0.824·53-s − 0.529·57-s + 1.28·61-s + 0.503·63-s − 0.248·65-s − 0.488·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{960} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.45732\)
\(L(\frac12)\) \(\approx\) \(2.45732\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
5 \( 1 - T \)
good7 \( 1 - 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−10.10989512091566913391448478748, −9.082506062594783472831133921500, −8.307525687782695431948900879973, −7.71052407307967077644420749342, −6.76895238282715006434981339956, −5.48025531888955534300415720953, −4.84143328212506167706839183620, −3.69936628544405197022333195268, −2.39041205999551187286748903960, −1.43294542786364492056174526795, 1.43294542786364492056174526795, 2.39041205999551187286748903960, 3.69936628544405197022333195268, 4.84143328212506167706839183620, 5.48025531888955534300415720953, 6.76895238282715006434981339956, 7.71052407307967077644420749342, 8.307525687782695431948900879973, 9.082506062594783472831133921500, 10.10989512091566913391448478748

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