Properties

Degree 2
Conductor $ 2^{6} \cdot 5 \cdot 7^{2} $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

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  =  0
Selberg data  =  $(2,\ 15680,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.586903684$
$L(\frac12)$  $\approx$  $1.586903684$
$L(\frac{3}{2})$   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

−16.09904132302926, −15.31825048053261, −14.77359666316792, −14.21890846352476, −14.06128649860745, −13.19705192949467, −12.64508865228612, −12.00034124665038, −11.60408932209105, −10.86574226765007, −10.45456235279372, −9.575770619721616, −9.174184662337894, −8.629421562945558, −8.039509248603006, −7.200102987600589, −6.517067982928951, −6.132128268936260, −5.380822005718795, −4.720053094879255, −3.894605182417161, −3.299527544003411, −2.249327599635781, −1.833897985553711, −0.5218814759142296, 0.5218814759142296, 1.833897985553711, 2.249327599635781, 3.299527544003411, 3.894605182417161, 4.720053094879255, 5.380822005718795, 6.132128268936260, 6.517067982928951, 7.200102987600589, 8.039509248603006, 8.629421562945558, 9.174184662337894, 9.575770619721616, 10.45456235279372, 10.86574226765007, 11.60408932209105, 12.00034124665038, 12.64508865228612, 13.19705192949467, 14.06128649860745, 14.21890846352476, 14.77359666316792, 15.31825048053261, 16.09904132302926

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