Properties

Degree 2
Conductor $ 2^{4} \cdot 5 \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  − 5-s − 4·7-s − 3·9-s + 4·11-s + 2·17-s + 4·19-s − 4·23-s + 25-s − 2·29-s − 8·31-s + 4·35-s − 6·37-s + 6·41-s + 8·43-s + 3·45-s + 4·47-s + 9·49-s + 6·53-s − 4·55-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 + ⋯
L(s)  = 1  − 0.447·5-s − 1.51·7-s − 9-s + 1.20·11-s + 0.485·17-s + 0.917·19-s − 0.834·23-s + 1/5·25-s − 0.371·29-s − 1.43·31-s + 0.676·35-s − 0.986·37-s + 0.937·41-s + 1.21·43-s + 0.447·45-s + 0.583·47-s + 9/7·49-s + 0.824·53-s − 0.539·55-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(13520\)    =    \(2^{4} \cdot 5 \cdot 13^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{13520} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 13520,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5,\;13\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;5,\;13\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + T \)
13 \( 1 \)
good3 \( 1 + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
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

−16.54194125082603, −15.94767470604637, −15.52150160695454, −14.71262254377032, −14.11244811158015, −13.94113553075322, −12.96370844798433, −12.49169142678052, −11.97876659594279, −11.49034711864024, −10.82398817026711, −10.12706777073587, −9.340970205768624, −9.200822048327765, −8.473263521240445, −7.558726925364482, −7.153300887002848, −6.341363734077218, −5.862548233810631, −5.297138454109411, −4.084167878031710, −3.624478177082778, −3.106125411953031, −2.195195187937987, −0.9542269717627968, 0, 0.9542269717627968, 2.195195187937987, 3.106125411953031, 3.624478177082778, 4.084167878031710, 5.297138454109411, 5.862548233810631, 6.341363734077218, 7.153300887002848, 7.558726925364482, 8.473263521240445, 9.200822048327765, 9.340970205768624, 10.12706777073587, 10.82398817026711, 11.49034711864024, 11.97876659594279, 12.49169142678052, 12.96370844798433, 13.94113553075322, 14.11244811158015, 14.71262254377032, 15.52150160695454, 15.94767470604637, 16.54194125082603

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