Properties

Degree 2
Conductor $ 2^{6} \cdot 5 \cdot 7^{2} $
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 − 3·9-s − 4·11-s − 2·13-s − 2·17-s + 4·19-s + 4·23-s + 25-s + 2·29-s + 8·31-s − 6·37-s + 6·41-s + 8·43-s − 3·45-s − 4·47-s − 6·53-s − 4·55-s − 4·59-s − 2·61-s − 2·65-s − 8·67-s + 6·73-s + 9·81-s − 16·83-s − 2·85-s + 6·89-s + 4·95-s + ⋯
L(s)  = 1  + 0.447·5-s − 9-s − 1.20·11-s − 0.554·13-s − 0.485·17-s + 0.917·19-s + 0.834·23-s + 1/5·25-s + 0.371·29-s + 1.43·31-s − 0.986·37-s + 0.937·41-s + 1.21·43-s − 0.447·45-s − 0.583·47-s − 0.824·53-s − 0.539·55-s − 0.520·59-s − 0.256·61-s − 0.248·65-s − 0.977·67-s + 0.702·73-s + 81-s − 1.75·83-s − 0.216·85-s + 0.635·89-s + 0.410·95-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(15680\)    =    \(2^{6} \cdot 5 \cdot 7^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{15680} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 15680,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5,\;7\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 - T \)
7 \( 1 \)
good3 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−16.13590281716230, −15.76224376939845, −15.23047571838505, −14.48024081080029, −14.06140789102038, −13.54264574005402, −13.00917450490071, −12.34843975596494, −11.83966126304957, −11.09419547877980, −10.70900422017625, −10.03709866176479, −9.453791619859658, −8.856488530961419, −8.242233035872451, −7.631213951510945, −7.044866053849803, −6.189805869335549, −5.707877423162941, −4.963730236233270, −4.620933028084727, −3.332399277853240, −2.787977987648449, −2.269996599488824, −1.059802197905912, 0, 1.059802197905912, 2.269996599488824, 2.787977987648449, 3.332399277853240, 4.620933028084727, 4.963730236233270, 5.707877423162941, 6.189805869335549, 7.044866053849803, 7.631213951510945, 8.242233035872451, 8.856488530961419, 9.453791619859658, 10.03709866176479, 10.70900422017625, 11.09419547877980, 11.83966126304957, 12.34843975596494, 13.00917450490071, 13.54264574005402, 14.06140789102038, 14.48024081080029, 15.23047571838505, 15.76224376939845, 16.13590281716230

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