Properties

Label 2-960-1.1-c3-0-38
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

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s − 5·5-s − 4·7-s + 9·9-s − 40·11-s + 90·13-s − 15·15-s − 70·17-s − 40·19-s − 12·21-s + 108·23-s + 25·25-s + 27·27-s − 166·29-s − 40·31-s − 120·33-s + 20·35-s + 130·37-s + 270·39-s − 310·41-s + 268·43-s − 45·45-s − 556·47-s − 327·49-s − 210·51-s + 370·53-s + 200·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s − 0.215·7-s + 1/3·9-s − 1.09·11-s + 1.92·13-s − 0.258·15-s − 0.998·17-s − 0.482·19-s − 0.124·21-s + 0.979·23-s + 1/5·25-s + 0.192·27-s − 1.06·29-s − 0.231·31-s − 0.633·33-s + 0.0965·35-s + 0.577·37-s + 1.10·39-s − 1.18·41-s + 0.950·43-s − 0.149·45-s − 1.72·47-s − 0.953·49-s − 0.576·51-s + 0.958·53-s + 0.490·55-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: \(1\)
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 \( 1 + 4 T + p^{3} T^{2} \)
11 \( 1 + 40 T + p^{3} T^{2} \)
13 \( 1 - 90 T + p^{3} T^{2} \)
17 \( 1 + 70 T + p^{3} T^{2} \)
19 \( 1 + 40 T + p^{3} T^{2} \)
23 \( 1 - 108 T + p^{3} T^{2} \)
29 \( 1 + 166 T + p^{3} T^{2} \)
31 \( 1 + 40 T + p^{3} T^{2} \)
37 \( 1 - 130 T + p^{3} T^{2} \)
41 \( 1 + 310 T + p^{3} T^{2} \)
43 \( 1 - 268 T + p^{3} T^{2} \)
47 \( 1 + 556 T + p^{3} T^{2} \)
53 \( 1 - 370 T + p^{3} T^{2} \)
59 \( 1 + 240 T + p^{3} T^{2} \)
61 \( 1 - 130 T + p^{3} T^{2} \)
67 \( 1 + 876 T + p^{3} T^{2} \)
71 \( 1 + 840 T + p^{3} T^{2} \)
73 \( 1 - 250 T + p^{3} T^{2} \)
79 \( 1 + 880 T + p^{3} T^{2} \)
83 \( 1 - 188 T + p^{3} T^{2} \)
89 \( 1 + 726 T + p^{3} T^{2} \)
97 \( 1 + 1550 T + p^{3} T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.945615649510797696707761954836, −8.522547823766320632033095461657, −7.66250472318992918521744703521, −6.74464722362504498654562508597, −5.82416685321950769710624579277, −4.64679331583814750684961567080, −3.69347220426552342538217971033, −2.83709946597990100342387899398, −1.53596338138111763827177591256, 0, 1.53596338138111763827177591256, 2.83709946597990100342387899398, 3.69347220426552342538217971033, 4.64679331583814750684961567080, 5.82416685321950769710624579277, 6.74464722362504498654562508597, 7.66250472318992918521744703521, 8.522547823766320632033095461657, 8.945615649510797696707761954836

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