Properties

Label 2-1008-1.1-c5-0-67
Degree $2$
Conductor $1008$
Sign $-1$
Analytic cond. $161.666$
Root an. cond. $12.7148$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 54·5-s − 49·7-s + 216·11-s + 998·13-s − 1.30e3·17-s − 884·19-s − 2.26e3·23-s − 209·25-s + 1.48e3·29-s − 8.36e3·31-s − 2.64e3·35-s − 4.71e3·37-s + 9.78e3·41-s − 1.94e4·43-s + 2.22e4·47-s + 2.40e3·49-s − 2.67e4·53-s + 1.16e4·55-s + 2.80e4·59-s − 3.88e4·61-s + 5.38e4·65-s − 2.39e4·67-s − 2.06e4·71-s + 290·73-s − 1.05e4·77-s + 9.95e4·79-s + 1.93e4·83-s + ⋯
L(s)  = 1  + 0.965·5-s − 0.377·7-s + 0.538·11-s + 1.63·13-s − 1.09·17-s − 0.561·19-s − 0.893·23-s − 0.0668·25-s + 0.327·29-s − 1.56·31-s − 0.365·35-s − 0.566·37-s + 0.909·41-s − 1.60·43-s + 1.46·47-s + 1/7·49-s − 1.31·53-s + 0.519·55-s + 1.05·59-s − 1.33·61-s + 1.58·65-s − 0.651·67-s − 0.485·71-s + 0.00636·73-s − 0.203·77-s + 1.79·79-s + 0.307·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-1$
Analytic conductor: \(161.666\)
Root analytic conductor: \(12.7148\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1008,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) 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 \( 1 \)
7 \( 1 + p^{2} T \)
good5 \( 1 - 54 T + p^{5} T^{2} \)
11 \( 1 - 216 T + p^{5} T^{2} \)
13 \( 1 - 998 T + p^{5} T^{2} \)
17 \( 1 + 1302 T + p^{5} T^{2} \)
19 \( 1 + 884 T + p^{5} T^{2} \)
23 \( 1 + 2268 T + p^{5} T^{2} \)
29 \( 1 - 1482 T + p^{5} T^{2} \)
31 \( 1 + 8360 T + p^{5} T^{2} \)
37 \( 1 + 4714 T + p^{5} T^{2} \)
41 \( 1 - 9786 T + p^{5} T^{2} \)
43 \( 1 + 452 p T + p^{5} T^{2} \)
47 \( 1 - 22200 T + p^{5} T^{2} \)
53 \( 1 + 26790 T + p^{5} T^{2} \)
59 \( 1 - 28092 T + p^{5} T^{2} \)
61 \( 1 + 38866 T + p^{5} T^{2} \)
67 \( 1 + 23948 T + p^{5} T^{2} \)
71 \( 1 + 20628 T + p^{5} T^{2} \)
73 \( 1 - 290 T + p^{5} T^{2} \)
79 \( 1 - 99544 T + p^{5} T^{2} \)
83 \( 1 - 19308 T + p^{5} T^{2} \)
89 \( 1 + 36390 T + p^{5} T^{2} \)
97 \( 1 + 79078 T + p^{5} T^{2} \)
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

−8.975841194989043934455910997280, −8.158907116723062076140859306815, −6.86406213221405836908371825764, −6.22361835429226416018918099370, −5.62943424280289244566848874695, −4.29678735839643011788345927947, −3.49515852852088336164371963628, −2.17221309357490688971466342634, −1.41197995458804473034080927152, 0, 1.41197995458804473034080927152, 2.17221309357490688971466342634, 3.49515852852088336164371963628, 4.29678735839643011788345927947, 5.62943424280289244566848874695, 6.22361835429226416018918099370, 6.86406213221405836908371825764, 8.158907116723062076140859306815, 8.975841194989043934455910997280

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