Properties

Degree $2$
Conductor $31680$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s − 4·7-s − 11-s + 4·13-s + 4·19-s + 6·23-s + 25-s − 6·29-s + 8·31-s − 4·35-s − 2·37-s − 6·41-s − 8·43-s − 6·47-s + 9·49-s − 6·53-s − 55-s − 12·59-s − 2·61-s + 4·65-s + 10·67-s + 12·71-s − 16·73-s + 4·77-s + 8·79-s − 6·89-s − 16·91-s + ⋯
L(s)  = 1  + 0.447·5-s − 1.51·7-s − 0.301·11-s + 1.10·13-s + 0.917·19-s + 1.25·23-s + 1/5·25-s − 1.11·29-s + 1.43·31-s − 0.676·35-s − 0.328·37-s − 0.937·41-s − 1.21·43-s − 0.875·47-s + 9/7·49-s − 0.824·53-s − 0.134·55-s − 1.56·59-s − 0.256·61-s + 0.496·65-s + 1.22·67-s + 1.42·71-s − 1.87·73-s + 0.455·77-s + 0.900·79-s − 0.635·89-s − 1.67·91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 31680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(31680\)    =    \(2^{6} \cdot 3^{2} \cdot 5 \cdot 11\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{31680} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 31680,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 - T \)
11 \( 1 + T \)
good7 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 16 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−15.39316702015709, −14.97447768714718, −14.07276351975540, −13.61584873062668, −13.26703480120546, −12.84210185993378, −12.28712968471581, −11.54328946420625, −11.10690378029400, −10.38609929946429, −9.959807498755737, −9.420051622861581, −9.013664404600185, −8.326849008146527, −7.703539175578376, −6.864068210875798, −6.548542835018125, −6.019938130190735, −5.319936183145588, −4.794148195713737, −3.771031466570881, −3.206210872458957, −2.906216023726088, −1.789546095593508, −1.018706115886145, 0, 1.018706115886145, 1.789546095593508, 2.906216023726088, 3.206210872458957, 3.771031466570881, 4.794148195713737, 5.319936183145588, 6.019938130190735, 6.548542835018125, 6.864068210875798, 7.703539175578376, 8.326849008146527, 9.013664404600185, 9.420051622861581, 9.959807498755737, 10.38609929946429, 11.10690378029400, 11.54328946420625, 12.28712968471581, 12.84210185993378, 13.26703480120546, 13.61584873062668, 14.07276351975540, 14.97447768714718, 15.39316702015709

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