Properties

Label 2-960-1.1-c3-0-29
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s − 5·5-s + 9·9-s + 4·11-s − 54·13-s + 15·15-s + 114·17-s + 44·19-s − 96·23-s + 25·25-s − 27·27-s − 134·29-s + 272·31-s − 12·33-s + 98·37-s + 162·39-s − 6·41-s + 12·43-s − 45·45-s + 200·47-s − 343·49-s − 342·51-s − 654·53-s − 20·55-s − 132·57-s + 36·59-s + 442·61-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 1/3·9-s + 0.109·11-s − 1.15·13-s + 0.258·15-s + 1.62·17-s + 0.531·19-s − 0.870·23-s + 1/5·25-s − 0.192·27-s − 0.858·29-s + 1.57·31-s − 0.0633·33-s + 0.435·37-s + 0.665·39-s − 0.0228·41-s + 0.0425·43-s − 0.149·45-s + 0.620·47-s − 49-s − 0.939·51-s − 1.69·53-s − 0.0490·55-s − 0.306·57-s + 0.0794·59-s + 0.927·61-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: \(1\)
Selberg data: \((2,\ 960,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 + p^{3} T^{2} \)
11 \( 1 - 4 T + p^{3} T^{2} \)
13 \( 1 + 54 T + p^{3} T^{2} \)
17 \( 1 - 114 T + p^{3} T^{2} \)
19 \( 1 - 44 T + p^{3} T^{2} \)
23 \( 1 + 96 T + p^{3} T^{2} \)
29 \( 1 + 134 T + p^{3} T^{2} \)
31 \( 1 - 272 T + p^{3} T^{2} \)
37 \( 1 - 98 T + p^{3} T^{2} \)
41 \( 1 + 6 T + p^{3} T^{2} \)
43 \( 1 - 12 T + p^{3} T^{2} \)
47 \( 1 - 200 T + p^{3} T^{2} \)
53 \( 1 + 654 T + p^{3} T^{2} \)
59 \( 1 - 36 T + p^{3} T^{2} \)
61 \( 1 - 442 T + p^{3} T^{2} \)
67 \( 1 + 188 T + p^{3} T^{2} \)
71 \( 1 - 632 T + p^{3} T^{2} \)
73 \( 1 + 390 T + p^{3} T^{2} \)
79 \( 1 + 688 T + p^{3} T^{2} \)
83 \( 1 - 1188 T + p^{3} T^{2} \)
89 \( 1 + 694 T + p^{3} T^{2} \)
97 \( 1 + 1726 T + p^{3} T^{2} \)
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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

−9.568505678983827912611308334142, −8.104228988919511163484185821745, −7.63694331579850094564635943622, −6.67171109196586027809219372304, −5.67441972389548955440720306995, −4.90095585015118614896590675873, −3.88380764458893199072478499973, −2.75987336849088362823397148840, −1.25983504831278892464004443697, 0, 1.25983504831278892464004443697, 2.75987336849088362823397148840, 3.88380764458893199072478499973, 4.90095585015118614896590675873, 5.67441972389548955440720306995, 6.67171109196586027809219372304, 7.63694331579850094564635943622, 8.104228988919511163484185821745, 9.568505678983827912611308334142

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