Properties

Degree 2
Conductor $ 2^{4} \cdot 3 \cdot 11 \cdot 13^{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  − 3-s − 2·5-s − 2·7-s + 9-s + 11-s + 2·15-s + 4·17-s − 6·19-s + 2·21-s − 25-s − 27-s − 8·29-s − 8·31-s − 33-s + 4·35-s − 10·37-s − 8·41-s + 2·43-s − 2·45-s − 8·47-s − 3·49-s − 4·51-s − 2·53-s − 2·55-s + 6·57-s + 12·59-s + 10·61-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.894·5-s − 0.755·7-s + 1/3·9-s + 0.301·11-s + 0.516·15-s + 0.970·17-s − 1.37·19-s + 0.436·21-s − 1/5·25-s − 0.192·27-s − 1.48·29-s − 1.43·31-s − 0.174·33-s + 0.676·35-s − 1.64·37-s − 1.24·41-s + 0.304·43-s − 0.298·45-s − 1.16·47-s − 3/7·49-s − 0.560·51-s − 0.274·53-s − 0.269·55-s + 0.794·57-s + 1.56·59-s + 1.28·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 89232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 89232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(89232\)    =    \(2^{4} \cdot 3 \cdot 11 \cdot 13^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{89232} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 89232,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;11,\;13\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;11,\;13\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
11 \( 1 - T \)
13 \( 1 \)
good5 \( 1 + 2 T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−14.25496466858539, −13.43578243790896, −13.01170753875220, −12.57815227298574, −12.22844281975795, −11.56711952164505, −11.30382652191058, −10.70038381909452, −10.20694311894262, −9.684294167689569, −9.209746791396241, −8.511284175067852, −8.103102288694718, −7.479797735605734, −6.902407113849555, −6.612537544988117, −5.916232717756071, −5.314927475946371, −4.937035031155925, −3.926771317051120, −3.694685653246139, −3.331625935898923, −2.146196650675746, −1.706679487599166, −0.5805066102372999, 0, 0.5805066102372999, 1.706679487599166, 2.146196650675746, 3.331625935898923, 3.694685653246139, 3.926771317051120, 4.937035031155925, 5.314927475946371, 5.916232717756071, 6.612537544988117, 6.902407113849555, 7.479797735605734, 8.103102288694718, 8.511284175067852, 9.209746791396241, 9.684294167689569, 10.20694311894262, 10.70038381909452, 11.30382652191058, 11.56711952164505, 12.22844281975795, 12.57815227298574, 13.01170753875220, 13.43578243790896, 14.25496466858539

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