Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 17 $
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 − 7-s − 6·13-s + 17-s − 4·19-s + 25-s + 2·29-s − 35-s − 6·37-s + 10·41-s − 4·43-s + 4·47-s + 49-s − 6·53-s + 4·59-s − 14·61-s − 6·65-s + 12·67-s + 12·71-s − 10·73-s + 12·79-s + 16·83-s + 85-s + 6·89-s + 6·91-s − 4·95-s − 18·97-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.377·7-s − 1.66·13-s + 0.242·17-s − 0.917·19-s + 1/5·25-s + 0.371·29-s − 0.169·35-s − 0.986·37-s + 1.56·41-s − 0.609·43-s + 0.583·47-s + 1/7·49-s − 0.824·53-s + 0.520·59-s − 1.79·61-s − 0.744·65-s + 1.46·67-s + 1.42·71-s − 1.17·73-s + 1.35·79-s + 1.75·83-s + 0.108·85-s + 0.635·89-s + 0.628·91-s − 0.410·95-s − 1.82·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 85680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(85680\)    =    \(2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 17\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{85680} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 85680,\ (\ :1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7,\;17\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;7,\;17\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 - T \)
7 \( 1 + T \)
17 \( 1 - T \)
good11 \( 1 + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−14.31087815837177, −13.70639889997101, −13.12371401369802, −12.59568385156156, −12.30980844170876, −11.86803883140079, −11.09734766586430, −10.59417594400369, −10.20645093283430, −9.633461888551464, −9.261578726193976, −8.770323740973464, −7.973797818879490, −7.641560630915864, −6.979763048640985, −6.477352314717081, −6.054724456522211, −5.208095085901543, −4.970832001928156, −4.252501427462021, −3.613722678445770, −2.857722037318546, −2.352794727616731, −1.816499235457574, −0.8032414793967418, 0, 0.8032414793967418, 1.816499235457574, 2.352794727616731, 2.857722037318546, 3.613722678445770, 4.252501427462021, 4.970832001928156, 5.208095085901543, 6.054724456522211, 6.477352314717081, 6.979763048640985, 7.641560630915864, 7.973797818879490, 8.770323740973464, 9.261578726193976, 9.633461888551464, 10.20645093283430, 10.59417594400369, 11.09734766586430, 11.86803883140079, 12.30980844170876, 12.59568385156156, 13.12371401369802, 13.70639889997101, 14.31087815837177

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