Properties

Label 2-960-1.1-c3-0-16
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 + 20·11-s + 58·13-s − 15·15-s − 70·17-s + 92·19-s − 36·21-s + 112·23-s + 25·25-s − 27·27-s − 66·29-s − 108·31-s − 60·33-s + 60·35-s + 58·37-s − 174·39-s + 66·41-s + 388·43-s + 45·45-s − 408·47-s − 199·49-s + 210·51-s − 474·53-s + 100·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 0.647·7-s + 1/3·9-s + 0.548·11-s + 1.23·13-s − 0.258·15-s − 0.998·17-s + 1.11·19-s − 0.374·21-s + 1.01·23-s + 1/5·25-s − 0.192·27-s − 0.422·29-s − 0.625·31-s − 0.316·33-s + 0.289·35-s + 0.257·37-s − 0.714·39-s + 0.251·41-s + 1.37·43-s + 0.149·45-s − 1.26·47-s − 0.580·49-s + 0.576·51-s − 1.22·53-s + 0.245·55-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

\(L(2)\) \(\approx\) \(2.317451911\)
\(L(\frac12)\) \(\approx\) \(2.317451911\)
\(L(\frac{5}{2})\) 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 \( 1 - 20 T + p^{3} T^{2} \)
13 \( 1 - 58 T + p^{3} T^{2} \)
17 \( 1 + 70 T + p^{3} T^{2} \)
19 \( 1 - 92 T + p^{3} T^{2} \)
23 \( 1 - 112 T + p^{3} T^{2} \)
29 \( 1 + 66 T + p^{3} T^{2} \)
31 \( 1 + 108 T + p^{3} T^{2} \)
37 \( 1 - 58 T + p^{3} T^{2} \)
41 \( 1 - 66 T + p^{3} T^{2} \)
43 \( 1 - 388 T + p^{3} T^{2} \)
47 \( 1 + 408 T + p^{3} T^{2} \)
53 \( 1 + 474 T + p^{3} T^{2} \)
59 \( 1 - 540 T + p^{3} T^{2} \)
61 \( 1 + 14 T + p^{3} T^{2} \)
67 \( 1 - 276 T + p^{3} T^{2} \)
71 \( 1 + 96 T + p^{3} T^{2} \)
73 \( 1 + 790 T + p^{3} T^{2} \)
79 \( 1 - 308 T + p^{3} T^{2} \)
83 \( 1 - 1036 T + p^{3} T^{2} \)
89 \( 1 - 1210 T + p^{3} T^{2} \)
97 \( 1 - 1426 T + p^{3} T^{2} \)
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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

−9.515786360333114641593391342273, −8.975076269207034907168621386349, −7.958101740704132812859808881935, −6.95722468874907160616417143672, −6.18462626064890717976700260767, −5.33780327463543922083551689695, −4.45853825130413822924713809879, −3.34609338950883992597918870534, −1.84177879209153487157950824601, −0.903729983872970344276875757282, 0.903729983872970344276875757282, 1.84177879209153487157950824601, 3.34609338950883992597918870534, 4.45853825130413822924713809879, 5.33780327463543922083551689695, 6.18462626064890717976700260767, 6.95722468874907160616417143672, 7.958101740704132812859808881935, 8.975076269207034907168621386349, 9.515786360333114641593391342273

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