Properties

Degree $2$
Conductor $10608$
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  + 3-s − 2·5-s + 9-s − 4·11-s + 13-s − 2·15-s + 17-s + 4·19-s − 25-s + 27-s − 2·29-s + 8·31-s − 4·33-s − 2·37-s + 39-s + 2·41-s + 4·43-s − 2·45-s − 8·47-s − 7·49-s + 51-s − 10·53-s + 8·55-s + 4·57-s − 4·59-s + 14·61-s − 2·65-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.894·5-s + 1/3·9-s − 1.20·11-s + 0.277·13-s − 0.516·15-s + 0.242·17-s + 0.917·19-s − 1/5·25-s + 0.192·27-s − 0.371·29-s + 1.43·31-s − 0.696·33-s − 0.328·37-s + 0.160·39-s + 0.312·41-s + 0.609·43-s − 0.298·45-s − 1.16·47-s − 49-s + 0.140·51-s − 1.37·53-s + 1.07·55-s + 0.529·57-s − 0.520·59-s + 1.79·61-s − 0.248·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10608\)    =    \(2^{4} \cdot 3 \cdot 13 \cdot 17\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{10608} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 10608,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
13 \( 1 - T \)
17 \( 1 - T \)
good5 \( 1 + 2 T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
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 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
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 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−16.61014588726586, −16.05807301178445, −15.66077655340819, −15.34131528858646, −14.50189349016622, −14.06228335148770, −13.36049626597087, −12.89202575254015, −12.28437361648148, −11.52596181852160, −11.20017647436948, −10.29410017376390, −9.872183175829691, −9.166664410080750, −8.314129172623915, −7.948822953231826, −7.530380404193677, −6.741207760872434, −5.910816029888285, −5.082467600945244, −4.512213701023178, −3.600395969072166, −3.108015208295201, −2.303139044636668, −1.179431515411365, 0, 1.179431515411365, 2.303139044636668, 3.108015208295201, 3.600395969072166, 4.512213701023178, 5.082467600945244, 5.910816029888285, 6.741207760872434, 7.530380404193677, 7.948822953231826, 8.314129172623915, 9.166664410080750, 9.872183175829691, 10.29410017376390, 11.20017647436948, 11.52596181852160, 12.28437361648148, 12.89202575254015, 13.36049626597087, 14.06228335148770, 14.50189349016622, 15.34131528858646, 15.66077655340819, 16.05807301178445, 16.61014588726586

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