Properties

Label 2-960-1.1-c3-0-18
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 + 8·7-s + 9·9-s + 4·11-s + 6·13-s + 15·15-s − 2·17-s − 16·19-s + 24·21-s + 60·23-s + 25·25-s + 27·27-s + 142·29-s + 176·31-s + 12·33-s + 40·35-s + 214·37-s + 18·39-s − 278·41-s − 68·43-s + 45·45-s − 116·47-s − 279·49-s − 6·51-s + 350·53-s + 20·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 0.431·7-s + 1/3·9-s + 0.109·11-s + 0.128·13-s + 0.258·15-s − 0.0285·17-s − 0.193·19-s + 0.249·21-s + 0.543·23-s + 1/5·25-s + 0.192·27-s + 0.909·29-s + 1.01·31-s + 0.0633·33-s + 0.193·35-s + 0.950·37-s + 0.0739·39-s − 1.05·41-s − 0.241·43-s + 0.149·45-s − 0.360·47-s − 0.813·49-s − 0.0164·51-s + 0.907·53-s + 0.0490·55-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

\(L(2)\) \(\approx\) \(3.226192849\)
\(L(\frac12)\) \(\approx\) \(3.226192849\)
\(L(\frac{5}{2})\) 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 \( 1 - 8 T + p^{3} T^{2} \)
11 \( 1 - 4 T + p^{3} T^{2} \)
13 \( 1 - 6 T + p^{3} T^{2} \)
17 \( 1 + 2 T + p^{3} T^{2} \)
19 \( 1 + 16 T + p^{3} T^{2} \)
23 \( 1 - 60 T + p^{3} T^{2} \)
29 \( 1 - 142 T + p^{3} T^{2} \)
31 \( 1 - 176 T + p^{3} T^{2} \)
37 \( 1 - 214 T + p^{3} T^{2} \)
41 \( 1 + 278 T + p^{3} T^{2} \)
43 \( 1 + 68 T + p^{3} T^{2} \)
47 \( 1 + 116 T + p^{3} T^{2} \)
53 \( 1 - 350 T + p^{3} T^{2} \)
59 \( 1 - 684 T + p^{3} T^{2} \)
61 \( 1 - 394 T + p^{3} T^{2} \)
67 \( 1 - 108 T + p^{3} T^{2} \)
71 \( 1 - 96 T + p^{3} T^{2} \)
73 \( 1 + 398 T + p^{3} T^{2} \)
79 \( 1 + 136 T + p^{3} T^{2} \)
83 \( 1 - 436 T + p^{3} T^{2} \)
89 \( 1 + 750 T + p^{3} T^{2} \)
97 \( 1 - 82 T + p^{3} T^{2} \)
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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

−9.698766522618382595080881591701, −8.661575080422118200623595417367, −8.203524931340164325122713610223, −7.09753764707136248028066210797, −6.32248953814704214578428981559, −5.19319332459082887871653917305, −4.32941272705009371353210576444, −3.14806221087549765034072285868, −2.15048893101352652539620862213, −0.989827037664358448552285888159, 0.989827037664358448552285888159, 2.15048893101352652539620862213, 3.14806221087549765034072285868, 4.32941272705009371353210576444, 5.19319332459082887871653917305, 6.32248953814704214578428981559, 7.09753764707136248028066210797, 8.203524931340164325122713610223, 8.661575080422118200623595417367, 9.698766522618382595080881591701

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