Properties

Label 2-960-1.1-c1-0-4
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 − 4·7-s + 9-s + 6·13-s + 15-s − 2·17-s + 4·19-s − 4·21-s + 8·23-s + 25-s + 27-s + 6·29-s − 4·35-s + 6·37-s + 6·39-s + 10·41-s − 4·43-s + 45-s − 8·47-s + 9·49-s − 2·51-s − 10·53-s + 4·57-s − 6·61-s − 4·63-s + 6·65-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s − 1.51·7-s + 1/3·9-s + 1.66·13-s + 0.258·15-s − 0.485·17-s + 0.917·19-s − 0.872·21-s + 1.66·23-s + 1/5·25-s + 0.192·27-s + 1.11·29-s − 0.676·35-s + 0.986·37-s + 0.960·39-s + 1.56·41-s − 0.609·43-s + 0.149·45-s − 1.16·47-s + 9/7·49-s − 0.280·51-s − 1.37·53-s + 0.529·57-s − 0.768·61-s − 0.503·63-s + 0.744·65-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

\(L(1)\) \(\approx\) \(1.977535250\)
\(L(\frac12)\) \(\approx\) \(1.977535250\)
\(L(\frac{3}{2})\) 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 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
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 \( 1 + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
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

−9.843986395238725085410957067541, −9.192233856919801710529869731712, −8.620274137521206455931441584726, −7.45166089807995498253197140252, −6.49211773175788373093258725474, −6.00871420537581383609527234257, −4.63883887090438359185158186412, −3.39015414339768891989414504202, −2.85965731186513405981248890470, −1.16849564057756810686392475492, 1.16849564057756810686392475492, 2.85965731186513405981248890470, 3.39015414339768891989414504202, 4.63883887090438359185158186412, 6.00871420537581383609527234257, 6.49211773175788373093258725474, 7.45166089807995498253197140252, 8.620274137521206455931441584726, 9.192233856919801710529869731712, 9.843986395238725085410957067541

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