Properties

Label 2-960-1.1-c1-0-5
Degree $2$
Conductor $960$
Sign $1$
Analytic cond. $7.66563$
Root an. cond. $2.76868$
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  + 3-s − 5-s + 9-s + 4·11-s + 2·13-s − 15-s + 2·17-s − 4·19-s + 25-s + 27-s + 2·29-s + 4·33-s + 10·37-s + 2·39-s + 10·41-s − 4·43-s − 45-s + 8·47-s − 7·49-s + 2·51-s + 10·53-s − 4·55-s − 4·57-s + 4·59-s + 2·61-s − 2·65-s − 12·67-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 1/3·9-s + 1.20·11-s + 0.554·13-s − 0.258·15-s + 0.485·17-s − 0.917·19-s + 1/5·25-s + 0.192·27-s + 0.371·29-s + 0.696·33-s + 1.64·37-s + 0.320·39-s + 1.56·41-s − 0.609·43-s − 0.149·45-s + 1.16·47-s − 49-s + 0.280·51-s + 1.37·53-s − 0.539·55-s − 0.529·57-s + 0.520·59-s + 0.256·61-s − 0.248·65-s − 1.46·67-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: $1$
Analytic conductor: \(7.66563\)
Root analytic conductor: \(2.76868\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.980751817\)
\(L(\frac12)\) \(\approx\) \(1.980751817\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−9.907104671360377962786893632766, −9.080341213166513123570518780286, −8.438902886313495638742074033009, −7.60125180356652570547951985370, −6.67875901406960725991694880905, −5.85130665991850657402735978605, −4.38871739502786245271092399731, −3.82349173075266898382654912582, −2.63252540175087701435373844808, −1.18815390030287753587945796282, 1.18815390030287753587945796282, 2.63252540175087701435373844808, 3.82349173075266898382654912582, 4.38871739502786245271092399731, 5.85130665991850657402735978605, 6.67875901406960725991694880905, 7.60125180356652570547951985370, 8.438902886313495638742074033009, 9.080341213166513123570518780286, 9.907104671360377962786893632766

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