Properties

Label 2-960-15.14-c2-0-49
Degree $2$
Conductor $960$
Sign $1$
Analytic cond. $26.1581$
Root an. cond. $5.11449$
Motivic weight $2$
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 + 9·9-s − 15·15-s − 14·17-s + 22·19-s + 34·23-s + 25·25-s + 27·27-s + 2·31-s − 45·45-s − 14·47-s + 49·49-s − 42·51-s + 86·53-s + 66·57-s + 118·61-s + 102·69-s + 75·75-s + 98·79-s + 81·81-s − 154·83-s + 70·85-s + 6·93-s − 110·95-s − 106·107-s + 22·109-s + ⋯
L(s)  = 1  + 3-s − 5-s + 9-s − 15-s − 0.823·17-s + 1.15·19-s + 1.47·23-s + 25-s + 27-s + 2/31·31-s − 45-s − 0.297·47-s + 49-s − 0.823·51-s + 1.62·53-s + 1.15·57-s + 1.93·61-s + 1.47·69-s + 75-s + 1.24·79-s + 81-s − 1.85·83-s + 0.823·85-s + 2/31·93-s − 1.15·95-s − 0.990·107-s + 0.201·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $1$
Analytic conductor: \(26.1581\)
Root analytic conductor: \(5.11449\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: $\chi_{960} (449, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.406862285\)
\(L(\frac12)\) \(\approx\) \(2.406862285\)
\(L(2)\) 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 \( ( 1 - p T )( 1 + p T ) \)
11 \( ( 1 - p T )( 1 + p T ) \)
13 \( ( 1 - p T )( 1 + p T ) \)
17 \( 1 + 14 T + p^{2} T^{2} \)
19 \( 1 - 22 T + p^{2} T^{2} \)
23 \( 1 - 34 T + p^{2} T^{2} \)
29 \( ( 1 - p T )( 1 + p T ) \)
31 \( 1 - 2 T + p^{2} T^{2} \)
37 \( ( 1 - p T )( 1 + p T ) \)
41 \( ( 1 - p T )( 1 + p T ) \)
43 \( ( 1 - p T )( 1 + p T ) \)
47 \( 1 + 14 T + p^{2} T^{2} \)
53 \( 1 - 86 T + p^{2} T^{2} \)
59 \( ( 1 - p T )( 1 + p T ) \)
61 \( 1 - 118 T + p^{2} T^{2} \)
67 \( ( 1 - p T )( 1 + p T ) \)
71 \( ( 1 - p T )( 1 + p T ) \)
73 \( ( 1 - p T )( 1 + p T ) \)
79 \( 1 - 98 T + p^{2} T^{2} \)
83 \( 1 + 154 T + p^{2} T^{2} \)
89 \( ( 1 - p T )( 1 + p T ) \)
97 \( ( 1 - p T )( 1 + p T ) \)
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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

−9.650656002337281562464649995460, −8.856335304749391979132766625134, −8.268705780996585607452380261920, −7.28880001385535669069123991695, −6.88117166867882854200098686257, −5.26939005940498076423117612453, −4.30022219795605510970309119756, −3.44770007742856349128488020872, −2.53297145717665903502010451478, −0.964153572142294797891098416898, 0.964153572142294797891098416898, 2.53297145717665903502010451478, 3.44770007742856349128488020872, 4.30022219795605510970309119756, 5.26939005940498076423117612453, 6.88117166867882854200098686257, 7.28880001385535669069123991695, 8.268705780996585607452380261920, 8.856335304749391979132766625134, 9.650656002337281562464649995460

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