Properties

Degree 2
Conductor $ 2^{3} \cdot 19 $
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  − 2·3-s − 5-s − 3·7-s + 9-s − 3·11-s − 4·13-s + 2·15-s + 5·17-s − 19-s + 6·21-s − 4·25-s + 4·27-s + 2·29-s + 8·31-s + 6·33-s + 3·35-s − 10·37-s + 8·39-s + 6·41-s − 7·43-s − 45-s − 9·47-s + 2·49-s − 10·51-s − 8·53-s + 3·55-s + 2·57-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.447·5-s − 1.13·7-s + 1/3·9-s − 0.904·11-s − 1.10·13-s + 0.516·15-s + 1.21·17-s − 0.229·19-s + 1.30·21-s − 4/5·25-s + 0.769·27-s + 0.371·29-s + 1.43·31-s + 1.04·33-s + 0.507·35-s − 1.64·37-s + 1.28·39-s + 0.937·41-s − 1.06·43-s − 0.149·45-s − 1.31·47-s + 2/7·49-s − 1.40·51-s − 1.09·53-s + 0.404·55-s + 0.264·57-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(152\)    =    \(2^{3} \cdot 19\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{152} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 152,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;19\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;19\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
19 \( 1 + T \)
good3 \( 1 + 2 T + p T^{2} \)
5 \( 1 + T + p T^{2} \)
7 \( 1 + 3 T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 5 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 7 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 + 8 T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 + 5 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 15 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 16 T + p T^{2} \)
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

−19.41797976911693, −19.12746120687565, −17.85411305430029, −17.13389222388040, −16.24470699058371, −15.70940911080945, −14.48379578967088, −13.18311090308489, −12.27242043590675, −11.72041328212866, −10.37952952672298, −9.829456006504791, −8.148422841499315, −6.977754927174973, −5.891527577804907, −4.851727589290649, −3.089150586355321, 0, 3.089150586355321, 4.851727589290649, 5.891527577804907, 6.977754927174973, 8.148422841499315, 9.829456006504791, 10.37952952672298, 11.72041328212866, 12.27242043590675, 13.18311090308489, 14.48379578967088, 15.70940911080945, 16.24470699058371, 17.13389222388040, 17.85411305430029, 19.12746120687565, 19.41797976911693

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