Properties

Degree 2
Conductor $ 2 \cdot 7 \cdot 11 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 4·5-s − 7-s − 8-s − 3·9-s + 4·10-s − 11-s + 2·13-s + 14-s + 16-s − 4·17-s + 3·18-s − 6·19-s − 4·20-s + 22-s + 4·23-s + 11·25-s − 2·26-s − 28-s − 2·29-s − 2·31-s − 32-s + 4·34-s + 4·35-s − 3·36-s + 10·37-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 1.78·5-s − 0.377·7-s − 0.353·8-s − 9-s + 1.26·10-s − 0.301·11-s + 0.554·13-s + 0.267·14-s + 1/4·16-s − 0.970·17-s + 0.707·18-s − 1.37·19-s − 0.894·20-s + 0.213·22-s + 0.834·23-s + 11/5·25-s − 0.392·26-s − 0.188·28-s − 0.371·29-s − 0.359·31-s − 0.176·32-s + 0.685·34-s + 0.676·35-s − 1/2·36-s + 1.64·37-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 154 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 154 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(154\)    =    \(2 \cdot 7 \cdot 11\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{154} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 154,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;7,\;11\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;7,\;11\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + T \)
7 \( 1 + T \)
11 \( 1 + T \)
good3 \( 1 + p T^{2} \)
5 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 4 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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

−19.83217111268814, −18.97559697183559, −18.20699274314285, −16.93070183382012, −16.35800458052419, −15.30011982678587, −14.91463756935541, −13.24884775406761, −12.22063198281703, −11.19149675055501, −10.82425456930031, −9.025591496790457, −8.383894826086909, −7.416203521394675, −6.228721970380122, −4.351071641285508, −2.983425967609043, 0, 2.983425967609043, 4.351071641285508, 6.228721970380122, 7.416203521394675, 8.383894826086909, 9.025591496790457, 10.82425456930031, 11.19149675055501, 12.22063198281703, 13.24884775406761, 14.91463756935541, 15.30011982678587, 16.35800458052419, 16.93070183382012, 18.20699274314285, 18.97559697183559, 19.83217111268814

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