Properties

Label 2-1008-1.1-c5-0-49
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  − 24·5-s − 49·7-s + 66·11-s + 98·13-s + 216·17-s + 340·19-s − 1.03e3·23-s − 2.54e3·25-s + 2.49e3·29-s + 7.04e3·31-s + 1.17e3·35-s − 1.22e4·37-s − 6.46e3·41-s + 1.54e4·43-s + 2.06e4·47-s + 2.40e3·49-s − 3.24e4·53-s − 1.58e3·55-s + 3.42e4·59-s + 3.56e4·61-s − 2.35e3·65-s − 1.26e4·67-s − 4.26e4·71-s + 3.37e4·73-s − 3.23e3·77-s + 8.51e4·79-s − 1.06e5·83-s + ⋯
L(s)  = 1  − 0.429·5-s − 0.377·7-s + 0.164·11-s + 0.160·13-s + 0.181·17-s + 0.216·19-s − 0.409·23-s − 0.815·25-s + 0.549·29-s + 1.31·31-s + 0.162·35-s − 1.46·37-s − 0.600·41-s + 1.27·43-s + 1.36·47-s + 1/7·49-s − 1.58·53-s − 0.0706·55-s + 1.27·59-s + 1.22·61-s − 0.0690·65-s − 0.345·67-s − 1.00·71-s + 0.740·73-s − 0.0621·77-s + 1.53·79-s − 1.70·83-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 \( 1 + 24 T + p^{5} T^{2} \)
11 \( 1 - 6 p T + p^{5} T^{2} \)
13 \( 1 - 98 T + p^{5} T^{2} \)
17 \( 1 - 216 T + p^{5} T^{2} \)
19 \( 1 - 340 T + p^{5} T^{2} \)
23 \( 1 + 1038 T + p^{5} T^{2} \)
29 \( 1 - 2490 T + p^{5} T^{2} \)
31 \( 1 - 7048 T + p^{5} T^{2} \)
37 \( 1 + 12238 T + p^{5} T^{2} \)
41 \( 1 + 6468 T + p^{5} T^{2} \)
43 \( 1 - 15412 T + p^{5} T^{2} \)
47 \( 1 - 20604 T + p^{5} T^{2} \)
53 \( 1 + 32490 T + p^{5} T^{2} \)
59 \( 1 - 34224 T + p^{5} T^{2} \)
61 \( 1 - 35654 T + p^{5} T^{2} \)
67 \( 1 + 12680 T + p^{5} T^{2} \)
71 \( 1 + 42642 T + p^{5} T^{2} \)
73 \( 1 - 33734 T + p^{5} T^{2} \)
79 \( 1 - 85108 T + p^{5} T^{2} \)
83 \( 1 + 106764 T + p^{5} T^{2} \)
89 \( 1 + 34884 T + p^{5} T^{2} \)
97 \( 1 - 18662 T + p^{5} T^{2} \)
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

−8.747561284291504676702214509006, −8.023586135061641197948728803496, −7.13841106656549616575453776184, −6.30429076092933604362391305404, −5.39584706482143657529289869226, −4.29181107740293824075708798459, −3.49672386614822657186412128462, −2.42562249114969190003236085476, −1.12574598006832431293799452181, 0, 1.12574598006832431293799452181, 2.42562249114969190003236085476, 3.49672386614822657186412128462, 4.29181107740293824075708798459, 5.39584706482143657529289869226, 6.30429076092933604362391305404, 7.13841106656549616575453776184, 8.023586135061641197948728803496, 8.747561284291504676702214509006

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