Properties

Degree 2
Conductor $ 3^{2} \cdot 5^{2} \cdot 11^{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  + 2-s − 4-s − 3·8-s − 2·13-s − 16-s − 2·17-s − 4·19-s − 2·26-s − 2·29-s + 5·32-s − 2·34-s + 10·37-s − 4·38-s + 10·41-s + 4·43-s + 8·47-s − 7·49-s + 2·52-s − 10·53-s − 2·58-s + 4·59-s + 2·61-s + 7·64-s − 12·67-s + 2·68-s + 8·71-s + 10·73-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s − 1.06·8-s − 0.554·13-s − 1/4·16-s − 0.485·17-s − 0.917·19-s − 0.392·26-s − 0.371·29-s + 0.883·32-s − 0.342·34-s + 1.64·37-s − 0.648·38-s + 1.56·41-s + 0.609·43-s + 1.16·47-s − 49-s + 0.277·52-s − 1.37·53-s − 0.262·58-s + 0.520·59-s + 0.256·61-s + 7/8·64-s − 1.46·67-s + 0.242·68-s + 0.949·71-s + 1.17·73-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 27225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 27225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(27225\)    =    \(3^{2} \cdot 5^{2} \cdot 11^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{27225} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 27225,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;5,\;11\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;5,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
11 \( 1 \)
good2 \( 1 - T + p T^{2} \)
7 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
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

−15.53784706591901, −14.70784370564101, −14.51642567340048, −14.01677526147472, −13.28992526700642, −12.82890823414186, −12.61406935511003, −11.89257894432477, −11.26252453717689, −10.81482595420928, −10.05341678007775, −9.390646211217299, −9.149136608245184, −8.377976054993821, −7.821750274991056, −7.195613149222092, −6.291150547443563, −6.041274260239905, −5.274538768684647, −4.547895985452651, −4.278394881285835, −3.529700056681857, −2.696149727714004, −2.179424371995501, −0.9326748827525501, 0, 0.9326748827525501, 2.179424371995501, 2.696149727714004, 3.529700056681857, 4.278394881285835, 4.547895985452651, 5.274538768684647, 6.041274260239905, 6.291150547443563, 7.195613149222092, 7.821750274991056, 8.377976054993821, 9.149136608245184, 9.390646211217299, 10.05341678007775, 10.81482595420928, 11.26252453717689, 11.89257894432477, 12.61406935511003, 12.82890823414186, 13.28992526700642, 14.01677526147472, 14.51642567340048, 14.70784370564101, 15.53784706591901

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