Properties

Label 2-960-1.1-c3-0-1
Degree $2$
Conductor $960$
Sign $1$
Analytic cond. $56.6418$
Root an. cond. $7.52607$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s + 5·5-s − 20·7-s + 9·9-s − 24·11-s − 74·13-s − 15·15-s + 54·17-s − 124·19-s + 60·21-s + 120·23-s + 25·25-s − 27·27-s + 78·29-s − 200·31-s + 72·33-s − 100·35-s + 70·37-s + 222·39-s + 330·41-s + 92·43-s + 45·45-s + 24·47-s + 57·49-s − 162·51-s − 450·53-s − 120·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s − 1.07·7-s + 1/3·9-s − 0.657·11-s − 1.57·13-s − 0.258·15-s + 0.770·17-s − 1.49·19-s + 0.623·21-s + 1.08·23-s + 1/5·25-s − 0.192·27-s + 0.499·29-s − 1.15·31-s + 0.379·33-s − 0.482·35-s + 0.311·37-s + 0.911·39-s + 1.25·41-s + 0.326·43-s + 0.149·45-s + 0.0744·47-s + 0.166·49-s − 0.444·51-s − 1.16·53-s − 0.294·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $1$
Analytic conductor: \(56.6418\)
Root analytic conductor: \(7.52607\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: $\chi_{960} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.9329722687\)
\(L(\frac12)\) \(\approx\) \(0.9329722687\)
\(L(\frac{5}{2})\) 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 \( 1 + p T \)
5 \( 1 - p T \)
good7 \( 1 + 20 T + p^{3} T^{2} \)
11 \( 1 + 24 T + p^{3} T^{2} \)
13 \( 1 + 74 T + p^{3} T^{2} \)
17 \( 1 - 54 T + p^{3} T^{2} \)
19 \( 1 + 124 T + p^{3} T^{2} \)
23 \( 1 - 120 T + p^{3} T^{2} \)
29 \( 1 - 78 T + p^{3} T^{2} \)
31 \( 1 + 200 T + p^{3} T^{2} \)
37 \( 1 - 70 T + p^{3} T^{2} \)
41 \( 1 - 330 T + p^{3} T^{2} \)
43 \( 1 - 92 T + p^{3} T^{2} \)
47 \( 1 - 24 T + p^{3} T^{2} \)
53 \( 1 + 450 T + p^{3} T^{2} \)
59 \( 1 - 24 T + p^{3} T^{2} \)
61 \( 1 - 322 T + p^{3} T^{2} \)
67 \( 1 + 196 T + p^{3} T^{2} \)
71 \( 1 - 288 T + p^{3} T^{2} \)
73 \( 1 + 430 T + p^{3} T^{2} \)
79 \( 1 - 520 T + p^{3} T^{2} \)
83 \( 1 - 156 T + p^{3} T^{2} \)
89 \( 1 - 1026 T + p^{3} T^{2} \)
97 \( 1 + 286 T + p^{3} T^{2} \)
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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

−9.733629609714883238692302085640, −9.072101551103016518387292291749, −7.78895620021114900221949216110, −6.99856254987306546810255407075, −6.20169051729730916277908063786, −5.35438382083019280139378531551, −4.50526366285400123004103083523, −3.13613700503052053857312340941, −2.19039060817134113378631151261, −0.50764496768865082402193622764, 0.50764496768865082402193622764, 2.19039060817134113378631151261, 3.13613700503052053857312340941, 4.50526366285400123004103083523, 5.35438382083019280139378531551, 6.20169051729730916277908063786, 6.99856254987306546810255407075, 7.78895620021114900221949216110, 9.072101551103016518387292291749, 9.733629609714883238692302085640

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