Properties

Label 2-960-1.1-c3-0-2
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 − 12·7-s + 9·9-s + 24·11-s − 38·13-s + 15·15-s − 6·17-s − 104·19-s + 36·21-s + 100·23-s + 25·25-s − 27·27-s − 230·29-s − 56·31-s − 72·33-s + 60·35-s − 190·37-s + 114·39-s + 202·41-s + 148·43-s − 45·45-s + 124·47-s − 199·49-s + 18·51-s − 206·53-s − 120·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s − 0.647·7-s + 1/3·9-s + 0.657·11-s − 0.810·13-s + 0.258·15-s − 0.0856·17-s − 1.25·19-s + 0.374·21-s + 0.906·23-s + 1/5·25-s − 0.192·27-s − 1.47·29-s − 0.324·31-s − 0.379·33-s + 0.289·35-s − 0.844·37-s + 0.468·39-s + 0.769·41-s + 0.524·43-s − 0.149·45-s + 0.384·47-s − 0.580·49-s + 0.0494·51-s − 0.533·53-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.9045537393\)
\(L(\frac12)\) \(\approx\) \(0.9045537393\)
\(L(\frac{5}{2})\) 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 \( 1 + 12 T + p^{3} T^{2} \)
11 \( 1 - 24 T + p^{3} T^{2} \)
13 \( 1 + 38 T + p^{3} T^{2} \)
17 \( 1 + 6 T + p^{3} T^{2} \)
19 \( 1 + 104 T + p^{3} T^{2} \)
23 \( 1 - 100 T + p^{3} T^{2} \)
29 \( 1 + 230 T + p^{3} T^{2} \)
31 \( 1 + 56 T + p^{3} T^{2} \)
37 \( 1 + 190 T + p^{3} T^{2} \)
41 \( 1 - 202 T + p^{3} T^{2} \)
43 \( 1 - 148 T + p^{3} T^{2} \)
47 \( 1 - 124 T + p^{3} T^{2} \)
53 \( 1 + 206 T + p^{3} T^{2} \)
59 \( 1 - 128 T + p^{3} T^{2} \)
61 \( 1 + 190 T + p^{3} T^{2} \)
67 \( 1 - 204 T + p^{3} T^{2} \)
71 \( 1 + 440 T + p^{3} T^{2} \)
73 \( 1 - 1210 T + p^{3} T^{2} \)
79 \( 1 - 816 T + p^{3} T^{2} \)
83 \( 1 - 1412 T + p^{3} T^{2} \)
89 \( 1 + 214 T + p^{3} T^{2} \)
97 \( 1 - 1202 T + p^{3} T^{2} \)
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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

−9.545267730913800701356567715691, −9.020013535251262613829214518534, −7.83066470396588684357774514298, −6.97954462847919751200317777552, −6.33587686899058210647324972565, −5.28779456208351688994386690313, −4.31723814942720429858120536701, −3.41613057107849039202310874677, −2.04559692150369116678735794181, −0.51047135976057947964302228633, 0.51047135976057947964302228633, 2.04559692150369116678735794181, 3.41613057107849039202310874677, 4.31723814942720429858120536701, 5.28779456208351688994386690313, 6.33587686899058210647324972565, 6.97954462847919751200317777552, 7.83066470396588684357774514298, 9.020013535251262613829214518534, 9.545267730913800701356567715691

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