Properties

Degree 2
Conductor $ 2^{4} \cdot 5^{2} \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  − 2·3-s + 9-s − 3·11-s + 13-s + 6·17-s + 19-s + 9·23-s + 4·27-s + 6·29-s − 8·31-s + 6·33-s + 7·37-s − 2·39-s + 3·41-s + 2·43-s + 9·47-s − 12·51-s − 9·53-s − 2·57-s + 8·61-s + 8·67-s − 18·69-s + 4·73-s + 10·79-s − 11·81-s − 12·87-s + 6·89-s + ⋯
L(s)  = 1  − 1.15·3-s + 1/3·9-s − 0.904·11-s + 0.277·13-s + 1.45·17-s + 0.229·19-s + 1.87·23-s + 0.769·27-s + 1.11·29-s − 1.43·31-s + 1.04·33-s + 1.15·37-s − 0.320·39-s + 0.468·41-s + 0.304·43-s + 1.31·47-s − 1.68·51-s − 1.23·53-s − 0.264·57-s + 1.02·61-s + 0.977·67-s − 2.16·69-s + 0.468·73-s + 1.12·79-s − 1.22·81-s − 1.28·87-s + 0.635·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(19600\)    =    \(2^{4} \cdot 5^{2} \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{19600} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 19600,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.516776732$
$L(\frac12)$  $\approx$  $1.516776732$
$L(\frac{3}{2})$   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 \( 1 \)
7 \( 1 \)
good3 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 - 9 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 7 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−15.98176170398575, −15.20149557583853, −14.56825003832735, −14.16175061072201, −13.34880556445110, −12.75280241698185, −12.47642096847361, −11.77927705365047, −11.21012864110233, −10.75076861102506, −10.39389743688369, −9.576763719972097, −9.077638899628656, −8.239136960965513, −7.678596546904705, −7.100686202277305, −6.384239301284739, −5.792043760261047, −5.172026631845243, −4.958801237759539, −3.903085791655132, −3.107309541982133, −2.480833721408374, −1.170418234593228, −0.6470261604898585, 0.6470261604898585, 1.170418234593228, 2.480833721408374, 3.107309541982133, 3.903085791655132, 4.958801237759539, 5.172026631845243, 5.792043760261047, 6.384239301284739, 7.100686202277305, 7.678596546904705, 8.239136960965513, 9.077638899628656, 9.576763719972097, 10.39389743688369, 10.75076861102506, 11.21012864110233, 11.77927705365047, 12.47642096847361, 12.75280241698185, 13.34880556445110, 14.16175061072201, 14.56825003832735, 15.20149557583853, 15.98176170398575

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