Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7^{2} \cdot 13 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 8-s + 13-s + 16-s − 6·17-s + 4·19-s + 26-s − 6·29-s + 4·31-s + 32-s − 6·34-s + 10·37-s + 4·38-s + 6·41-s − 8·43-s + 52-s − 6·53-s − 6·58-s − 12·59-s − 14·61-s + 4·62-s + 64-s + 4·67-s − 6·68-s + 2·73-s + 10·74-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.353·8-s + 0.277·13-s + 1/4·16-s − 1.45·17-s + 0.917·19-s + 0.196·26-s − 1.11·29-s + 0.718·31-s + 0.176·32-s − 1.02·34-s + 1.64·37-s + 0.648·38-s + 0.937·41-s − 1.21·43-s + 0.138·52-s − 0.824·53-s − 0.787·58-s − 1.56·59-s − 1.79·61-s + 0.508·62-s + 1/8·64-s + 0.488·67-s − 0.727·68-s + 0.234·73-s + 1.16·74-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(286650\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7^{2} \cdot 13\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{286650} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 286650,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7,\;13\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;7,\;13\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
13 \( 1 - T \)
good11 \( 1 + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−13.05957124006076, −12.60385517215265, −11.97566052039350, −11.67152234506112, −11.10585449256841, −10.88666432723297, −10.35517366835805, −9.659538278108796, −9.211148070048841, −9.023589757774243, −8.060764042754454, −7.844747884626273, −7.389798903488248, −6.592814152436214, −6.417886021101832, −5.906082230100062, −5.305946643489700, −4.731187208037989, −4.433072886077378, −3.818633726375058, −3.248817210449974, −2.763278852947059, −2.147476612201964, −1.586161746348575, −0.8655793790134818, 0, 0.8655793790134818, 1.586161746348575, 2.147476612201964, 2.763278852947059, 3.248817210449974, 3.818633726375058, 4.433072886077378, 4.731187208037989, 5.305946643489700, 5.906082230100062, 6.417886021101832, 6.592814152436214, 7.389798903488248, 7.844747884626273, 8.060764042754454, 9.023589757774243, 9.211148070048841, 9.659538278108796, 10.35517366835805, 10.88666432723297, 11.10585449256841, 11.67152234506112, 11.97566052039350, 12.60385517215265, 13.05957124006076

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