Properties

Degree 2
Conductor $ 2^{4} \cdot 5^{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  − 4·7-s − 3·9-s − 4·11-s + 2·13-s − 2·17-s − 4·19-s + 4·23-s − 2·29-s + 8·31-s − 6·37-s − 6·41-s − 8·43-s + 4·47-s + 9·49-s − 6·53-s + 4·59-s − 2·61-s + 12·63-s + 8·67-s + 6·73-s + 16·77-s + 9·81-s − 16·83-s − 6·89-s − 8·91-s + 14·97-s + 12·99-s + ⋯
L(s)  = 1  − 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 − 0.371·29-s + 1.43·31-s − 0.986·37-s − 0.937·41-s − 1.21·43-s + 0.583·47-s + 9/7·49-s − 0.824·53-s + 0.520·59-s − 0.256·61-s + 1.51·63-s + 0.977·67-s + 0.702·73-s + 1.82·77-s + 81-s − 1.75·83-s − 0.635·89-s − 0.838·91-s + 1.42·97-s + 1.20·99-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(400\)    =    \(2^{4} \cdot 5^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{400} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 400,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

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

−19.67257407654709, −19.03113358189415, −18.37811935550994, −17.24725174892224, −16.74182895973427, −15.66001700224663, −15.39230238170346, −14.08369338722871, −13.24151287630455, −12.80843771617490, −11.68418131346840, −10.71447606291806, −10.01394722702472, −8.913111768905373, −8.231140090828504, −6.849262264669355, −6.128318729933684, −5.049619048227527, −3.487714330973268, −2.591370302894794, 0, 2.591370302894794, 3.487714330973268, 5.049619048227527, 6.128318729933684, 6.849262264669355, 8.231140090828504, 8.913111768905373, 10.01394722702472, 10.71447606291806, 11.68418131346840, 12.80843771617490, 13.24151287630455, 14.08369338722871, 15.39230238170346, 15.66001700224663, 16.74182895973427, 17.24725174892224, 18.37811935550994, 19.03113358189415, 19.67257407654709

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