Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{2} \cdot 431 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·5-s − 4·7-s − 5·11-s + 4·13-s + 6·17-s + 7·19-s − 3·23-s + 4·25-s − 3·29-s − 4·31-s + 12·35-s + 8·37-s + 6·41-s + 8·43-s + 6·47-s + 9·49-s − 53-s + 15·55-s − 9·59-s + 2·61-s − 12·65-s − 10·67-s + 8·71-s − 6·73-s + 20·77-s − 16·79-s − 16·83-s + ⋯
L(s)  = 1  − 1.34·5-s − 1.51·7-s − 1.50·11-s + 1.10·13-s + 1.45·17-s + 1.60·19-s − 0.625·23-s + 4/5·25-s − 0.557·29-s − 0.718·31-s + 2.02·35-s + 1.31·37-s + 0.937·41-s + 1.21·43-s + 0.875·47-s + 9/7·49-s − 0.137·53-s + 2.02·55-s − 1.17·59-s + 0.256·61-s − 1.48·65-s − 1.22·67-s + 0.949·71-s − 0.702·73-s + 2.27·77-s − 1.80·79-s − 1.75·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 124128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124128 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(124128\)    =    \(2^{5} \cdot 3^{2} \cdot 431\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{124128} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 124128,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;431\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;431\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
431 \( 1 + T \)
good5 \( 1 + 3 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + T + p T^{2} \)
59 \( 1 + 9 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 5 T + p T^{2} \)
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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

−13.75560156851781, −13.11994957705063, −12.81421941336372, −12.38153994295307, −11.90363998685781, −11.38459988988210, −10.85501539967603, −10.46597821229584, −9.789046948729492, −9.513268244102572, −8.925497509389631, −8.167667206363701, −7.806158493965557, −7.400132483784809, −7.116553020294082, −6.059213929170508, −5.782445939415133, −5.441124591361152, −4.446476582732594, −3.962939127808348, −3.431762592423383, −3.010028187721145, −2.611928006032861, −1.348719751662048, −0.6779820667308241, 0, 0.6779820667308241, 1.348719751662048, 2.611928006032861, 3.010028187721145, 3.431762592423383, 3.962939127808348, 4.446476582732594, 5.441124591361152, 5.782445939415133, 6.059213929170508, 7.116553020294082, 7.400132483784809, 7.806158493965557, 8.167667206363701, 8.925497509389631, 9.513268244102572, 9.789046948729492, 10.46597821229584, 10.85501539967603, 11.38459988988210, 11.90363998685781, 12.38153994295307, 12.81421941336372, 13.11994957705063, 13.75560156851781

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