Properties

Label 2-960-1.1-c3-0-12
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 − 4·7-s + 9·9-s + 72·11-s + 6·13-s − 15·15-s + 38·17-s + 52·19-s + 12·21-s − 152·23-s + 25·25-s − 27·27-s + 78·29-s − 120·31-s − 216·33-s − 20·35-s + 150·37-s − 18·39-s + 362·41-s − 484·43-s + 45·45-s − 280·47-s − 327·49-s − 114·51-s + 670·53-s + 360·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s − 0.215·7-s + 1/3·9-s + 1.97·11-s + 0.128·13-s − 0.258·15-s + 0.542·17-s + 0.627·19-s + 0.124·21-s − 1.37·23-s + 1/5·25-s − 0.192·27-s + 0.499·29-s − 0.695·31-s − 1.13·33-s − 0.0965·35-s + 0.666·37-s − 0.0739·39-s + 1.37·41-s − 1.71·43-s + 0.149·45-s − 0.868·47-s − 0.953·49-s − 0.313·51-s + 1.73·53-s + 0.882·55-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

\(L(2)\) \(\approx\) \(2.122730611\)
\(L(\frac12)\) \(\approx\) \(2.122730611\)
\(L(\frac{5}{2})\) 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 \( 1 + 4 T + p^{3} T^{2} \)
11 \( 1 - 72 T + p^{3} T^{2} \)
13 \( 1 - 6 T + p^{3} T^{2} \)
17 \( 1 - 38 T + p^{3} T^{2} \)
19 \( 1 - 52 T + p^{3} T^{2} \)
23 \( 1 + 152 T + p^{3} T^{2} \)
29 \( 1 - 78 T + p^{3} T^{2} \)
31 \( 1 + 120 T + p^{3} T^{2} \)
37 \( 1 - 150 T + p^{3} T^{2} \)
41 \( 1 - 362 T + p^{3} T^{2} \)
43 \( 1 + 484 T + p^{3} T^{2} \)
47 \( 1 + 280 T + p^{3} T^{2} \)
53 \( 1 - 670 T + p^{3} T^{2} \)
59 \( 1 - 696 T + p^{3} T^{2} \)
61 \( 1 + 222 T + p^{3} T^{2} \)
67 \( 1 + 4 T + p^{3} T^{2} \)
71 \( 1 + 96 T + p^{3} T^{2} \)
73 \( 1 - 178 T + p^{3} T^{2} \)
79 \( 1 - 8 p T + p^{3} T^{2} \)
83 \( 1 + 612 T + p^{3} T^{2} \)
89 \( 1 - 994 T + p^{3} T^{2} \)
97 \( 1 - 1634 T + p^{3} T^{2} \)
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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

−9.694195323879243096835004531647, −9.016156973049324375378914430572, −7.938802612998934683464376746474, −6.83462586523641184516561892171, −6.25818407682473594512947975548, −5.45805436944493450827973888879, −4.27621979253684225048519156263, −3.44236155542896746714344038420, −1.85585517837371393446113207264, −0.853122488878997241797130935729, 0.853122488878997241797130935729, 1.85585517837371393446113207264, 3.44236155542896746714344038420, 4.27621979253684225048519156263, 5.45805436944493450827973888879, 6.25818407682473594512947975548, 6.83462586523641184516561892171, 7.938802612998934683464376746474, 9.016156973049324375378914430572, 9.694195323879243096835004531647

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