Properties

Degree 2
Conductor $ 2^{4} \cdot 5 $
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 + 4·7-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 + 4·35-s + 6·37-s − 6·41-s + 8·43-s − 3·45-s − 4·47-s + 9·49-s + 6·53-s − 4·55-s + 4·59-s − 2·61-s − 12·63-s − 2·65-s − 8·67-s − 6·73-s − 16·77-s + ⋯
L(s)  = 1  + 0.447·5-s + 1.51·7-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 + 0.676·35-s + 0.986·37-s − 0.937·41-s + 1.21·43-s − 0.447·45-s − 0.583·47-s + 9/7·49-s + 0.824·53-s − 0.539·55-s + 0.520·59-s − 0.256·61-s − 1.51·63-s − 0.248·65-s − 0.977·67-s − 0.702·73-s − 1.82·77-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(80\)    =    \(2^{4} \cdot 5\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{80} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 80,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.009452909$
$L(\frac12)$  $\approx$  $1.009452909$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;5\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
5 \( 1 - T \)
good3 \( 1 + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
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

−19.32676481754105, −18.12898812409444, −17.52852234279302, −16.62864766814362, −15.16357915995616, −14.42381862707542, −13.50406820695137, −12.13050487476074, −11.11321493608158, −10.12246838996141, −8.521590142691915, −7.753274429927942, −5.849067365429761, −4.769056611195523, −2.399619781816413, 2.399619781816413, 4.769056611195523, 5.849067365429761, 7.753274429927942, 8.521590142691915, 10.12246838996141, 11.11321493608158, 12.13050487476074, 13.50406820695137, 14.42381862707542, 15.16357915995616, 16.62864766814362, 17.52852234279302, 18.12898812409444, 19.32676481754105

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