Properties

Degree 2
Conductor $ 3^{2} \cdot 5^{2} \cdot 7^{2} \cdot 11 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 4-s + 3·8-s + 11-s + 6·13-s − 16-s − 2·17-s − 4·19-s − 22-s − 6·26-s + 2·29-s − 8·31-s − 5·32-s + 2·34-s − 6·37-s + 4·38-s + 10·41-s + 4·43-s − 44-s + 8·47-s − 6·52-s + 6·53-s − 2·58-s + 4·59-s + 10·61-s + 8·62-s + 7·64-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s + 1.06·8-s + 0.301·11-s + 1.66·13-s − 1/4·16-s − 0.485·17-s − 0.917·19-s − 0.213·22-s − 1.17·26-s + 0.371·29-s − 1.43·31-s − 0.883·32-s + 0.342·34-s − 0.986·37-s + 0.648·38-s + 1.56·41-s + 0.609·43-s − 0.150·44-s + 1.16·47-s − 0.832·52-s + 0.824·53-s − 0.262·58-s + 0.520·59-s + 1.28·61-s + 1.01·62-s + 7/8·64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 121275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(121275\)    =    \(3^{2} \cdot 5^{2} \cdot 7^{2} \cdot 11\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{121275} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 121275,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(1.818418377\)
\(L(\frac12)\)  \(\approx\)  \(1.818418377\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;5,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
11 \( 1 - T \)
good2 \( 1 + T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
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 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−13.45544551807400, −13.17213604225243, −12.64821479610686, −12.19245336345861, −11.39727876112431, −10.87010685812935, −10.77388391581427, −10.19358083519156, −9.438289502642868, −9.132711849825662, −8.656239259152455, −8.355056065044027, −7.781311489559199, −7.104330792933559, −6.725197621858931, −5.984335789820791, −5.609066110256673, −4.912500489152348, −4.201910401765778, −3.863111214760269, −3.410251839877396, −2.252568169341731, −1.913252354751092, −0.9448968944502870, −0.6018912710863843, 0.6018912710863843, 0.9448968944502870, 1.913252354751092, 2.252568169341731, 3.410251839877396, 3.863111214760269, 4.201910401765778, 4.912500489152348, 5.609066110256673, 5.984335789820791, 6.725197621858931, 7.104330792933559, 7.781311489559199, 8.355056065044027, 8.656239259152455, 9.132711849825662, 9.438289502642868, 10.19358083519156, 10.77388391581427, 10.87010685812935, 11.39727876112431, 12.19245336345861, 12.64821479610686, 13.17213604225243, 13.45544551807400

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