Properties

Degree 2
Conductor $ 2^{4} \cdot 5 $
Sign $1$
Motivic weight 0
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 9-s + 25-s + 2·29-s − 2·41-s + 45-s − 49-s − 2·61-s + 81-s + 2·89-s + 2·101-s − 2·109-s + ⋯
L(s)  = 1  − 5-s − 9-s + 25-s + 2·29-s − 2·41-s + 45-s − 49-s − 2·61-s + 81-s + 2·89-s + 2·101-s − 2·109-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(80\)    =    \(2^{4} \cdot 5\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{80} (79, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 80,\ (\ :0),\ 1)$
$L(\frac{1}{2})$  $\approx$  $0.4643475584$
$L(\frac12)$  $\approx$  $0.4643475584$
$L(1)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5\}$, \(F_p\) is a polynomial of degree 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 \( 1 + T^{2} \)
7 \( 1 + T^{2} \)
11 \( ( 1 - T )( 1 + T ) \)
13 \( ( 1 - T )( 1 + T ) \)
17 \( ( 1 - T )( 1 + T ) \)
19 \( ( 1 - T )( 1 + T ) \)
23 \( 1 + T^{2} \)
29 \( ( 1 - T )^{2} \)
31 \( ( 1 - T )( 1 + T ) \)
37 \( ( 1 - T )( 1 + T ) \)
41 \( ( 1 + T )^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + T^{2} \)
53 \( ( 1 - T )( 1 + T ) \)
59 \( ( 1 - T )( 1 + T ) \)
61 \( ( 1 + T )^{2} \)
67 \( 1 + T^{2} \)
71 \( ( 1 - T )( 1 + T ) \)
73 \( ( 1 - T )( 1 + T ) \)
79 \( ( 1 - T )( 1 + T ) \)
83 \( 1 + T^{2} \)
89 \( ( 1 - T )^{2} \)
97 \( ( 1 - T )( 1 + T ) \)
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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

−14.71004762413275441512730195253, −13.69240792583054166179238644349, −12.27030224826286889685238700312, −11.54177841888707105180080511420, −10.41708110873279474671747278381, −8.820916987388348246449123416127, −7.935729000773875944090370436264, −6.49941746742761161543060822888, −4.84156760187899842160701524540, −3.18874463093794795645386180893, 3.18874463093794795645386180893, 4.84156760187899842160701524540, 6.49941746742761161543060822888, 7.935729000773875944090370436264, 8.820916987388348246449123416127, 10.41708110873279474671747278381, 11.54177841888707105180080511420, 12.27030224826286889685238700312, 13.69240792583054166179238644349, 14.71004762413275441512730195253

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