Properties

Degree $2$
Conductor $15680$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 3·9-s − 4·11-s − 6·13-s − 2·17-s + 25-s − 6·29-s − 8·31-s + 10·37-s − 2·41-s − 4·43-s + 3·45-s − 8·47-s + 2·53-s + 4·55-s − 8·59-s − 14·61-s + 6·65-s + 12·67-s − 16·71-s − 2·73-s − 8·79-s + 9·81-s + 8·83-s + 2·85-s − 10·89-s − 2·97-s + ⋯
L(s)  = 1  − 0.447·5-s − 9-s − 1.20·11-s − 1.66·13-s − 0.485·17-s + 1/5·25-s − 1.11·29-s − 1.43·31-s + 1.64·37-s − 0.312·41-s − 0.609·43-s + 0.447·45-s − 1.16·47-s + 0.274·53-s + 0.539·55-s − 1.04·59-s − 1.79·61-s + 0.744·65-s + 1.46·67-s − 1.89·71-s − 0.234·73-s − 0.900·79-s + 81-s + 0.878·83-s + 0.216·85-s − 1.05·89-s − 0.203·97-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

Degree: \(2\)
Conductor: \(15680\)    =    \(2^{6} \cdot 5 \cdot 7^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{15680} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((2,\ 15680,\ (\ :1/2),\ 1)\)

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 \( 1 \)
5 \( 1 + T \)
7 \( 1 \)
good3 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−16.56739417988620, −16.10570336370448, −15.22555519722006, −14.90411463684232, −14.55600118312933, −13.75999366610562, −13.09290548407192, −12.73998338162403, −12.02375700962658, −11.50718729619303, −10.94072379012947, −10.47802286686198, −9.582269129704764, −9.292696971976882, −8.424906545652173, −7.820828551097045, −7.485835514320124, −6.756087765363150, −5.870924518987700, −5.296648544264432, −4.792144891504565, −4.002244358002195, −2.995893500312555, −2.635901560358558, −1.745060466171496, 0, 0, 1.745060466171496, 2.635901560358558, 2.995893500312555, 4.002244358002195, 4.792144891504565, 5.296648544264432, 5.870924518987700, 6.756087765363150, 7.485835514320124, 7.820828551097045, 8.424906545652173, 9.292696971976882, 9.582269129704764, 10.47802286686198, 10.94072379012947, 11.50718729619303, 12.02375700962658, 12.73998338162403, 13.09290548407192, 13.75999366610562, 14.55600118312933, 14.90411463684232, 15.22555519722006, 16.10570336370448, 16.56739417988620

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