Properties

Degree $2$
Conductor $6720$
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  + 3-s + 5-s − 7-s + 9-s − 2·13-s + 15-s − 6·17-s − 4·19-s − 21-s + 25-s + 27-s + 6·29-s + 4·31-s − 35-s − 2·37-s − 2·39-s + 6·41-s + 8·43-s + 45-s + 12·47-s + 49-s − 6·51-s − 6·53-s − 4·57-s − 12·59-s − 2·61-s − 63-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s − 0.554·13-s + 0.258·15-s − 1.45·17-s − 0.917·19-s − 0.218·21-s + 1/5·25-s + 0.192·27-s + 1.11·29-s + 0.718·31-s − 0.169·35-s − 0.328·37-s − 0.320·39-s + 0.937·41-s + 1.21·43-s + 0.149·45-s + 1.75·47-s + 1/7·49-s − 0.840·51-s − 0.824·53-s − 0.529·57-s − 1.56·59-s − 0.256·61-s − 0.125·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6720\)    =    \(2^{6} \cdot 3 \cdot 5 \cdot 7\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{6720} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6720,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.324935714\)
\(L(\frac12)\) \(\approx\) \(2.324935714\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
5 \( 1 - T \)
7 \( 1 + T \)
good11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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

−17.29804018341821, −16.79188526829853, −15.78652408450402, −15.59368461519773, −14.93577418892087, −14.02873456040712, −13.92208788163321, −13.08856897294451, −12.54831351996862, −12.07557575832637, −10.96305652861608, −10.63895935873986, −9.877248384792667, −9.142700696077423, −8.866363151191319, −7.978515145485646, −7.331088549024098, −6.417687723170163, −6.201584093645790, −4.927435423497789, −4.443255773306374, −3.552477487319750, −2.484398226003008, −2.185671064238350, −0.7388382541703345, 0.7388382541703345, 2.185671064238350, 2.484398226003008, 3.552477487319750, 4.443255773306374, 4.927435423497789, 6.201584093645790, 6.417687723170163, 7.331088549024098, 7.978515145485646, 8.866363151191319, 9.142700696077423, 9.877248384792667, 10.63895935873986, 10.96305652861608, 12.07557575832637, 12.54831351996862, 13.08856897294451, 13.92208788163321, 14.02873456040712, 14.93577418892087, 15.59368461519773, 15.78652408450402, 16.79188526829853, 17.29804018341821

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