Properties

Label 2-1008-1.1-c5-0-24
Degree $2$
Conductor $1008$
Sign $1$
Analytic cond. $161.666$
Root an. cond. $12.7148$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 57.2·5-s − 49·7-s + 515.·11-s − 670·13-s − 973.·17-s − 284·19-s + 1.77e3·23-s + 151·25-s + 6.86e3·29-s − 1.53e3·31-s − 2.80e3·35-s − 1.51e4·37-s + 5.32e3·41-s + 1.09e4·43-s + 1.93e4·47-s + 2.40e3·49-s + 2.34e4·53-s + 2.94e4·55-s + 2.36e4·59-s − 1.46e4·61-s − 3.83e4·65-s + 3.66e4·67-s − 6.79e4·71-s − 5.48e4·73-s − 2.52e4·77-s + 3.17e4·79-s + 7.41e4·83-s + ⋯
L(s)  = 1  + 1.02·5-s − 0.377·7-s + 1.28·11-s − 1.09·13-s − 0.816·17-s − 0.180·19-s + 0.699·23-s + 0.0483·25-s + 1.51·29-s − 0.286·31-s − 0.386·35-s − 1.81·37-s + 0.494·41-s + 0.906·43-s + 1.27·47-s + 0.142·49-s + 1.14·53-s + 1.31·55-s + 0.886·59-s − 0.502·61-s − 1.12·65-s + 0.996·67-s − 1.59·71-s − 1.20·73-s − 0.485·77-s + 0.572·79-s + 1.18·83-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: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(2.686217015\)
\(L(\frac12)\) \(\approx\) \(2.686217015\)
\(L(\frac{7}{2})\) 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 \( 1 + 49T \)
good5 \( 1 - 57.2T + 3.12e3T^{2} \)
11 \( 1 - 515.T + 1.61e5T^{2} \)
13 \( 1 + 670T + 3.71e5T^{2} \)
17 \( 1 + 973.T + 1.41e6T^{2} \)
19 \( 1 + 284T + 2.47e6T^{2} \)
23 \( 1 - 1.77e3T + 6.43e6T^{2} \)
29 \( 1 - 6.86e3T + 2.05e7T^{2} \)
31 \( 1 + 1.53e3T + 2.86e7T^{2} \)
37 \( 1 + 1.51e4T + 6.93e7T^{2} \)
41 \( 1 - 5.32e3T + 1.15e8T^{2} \)
43 \( 1 - 1.09e4T + 1.47e8T^{2} \)
47 \( 1 - 1.93e4T + 2.29e8T^{2} \)
53 \( 1 - 2.34e4T + 4.18e8T^{2} \)
59 \( 1 - 2.36e4T + 7.14e8T^{2} \)
61 \( 1 + 1.46e4T + 8.44e8T^{2} \)
67 \( 1 - 3.66e4T + 1.35e9T^{2} \)
71 \( 1 + 6.79e4T + 1.80e9T^{2} \)
73 \( 1 + 5.48e4T + 2.07e9T^{2} \)
79 \( 1 - 3.17e4T + 3.07e9T^{2} \)
83 \( 1 - 7.41e4T + 3.93e9T^{2} \)
89 \( 1 + 858.T + 5.58e9T^{2} \)
97 \( 1 - 1.41e4T + 8.58e9T^{2} \)
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

−9.136405585163481890600679569711, −8.790238837846345152569362530670, −7.30638868497843428313417528525, −6.68523229061371586426743717277, −5.91094702021289906999846783998, −4.93267325054346283417616782638, −3.97626141773580400613243344731, −2.72325357697640913470240784477, −1.88266571798756806422390956403, −0.71332634715229048141580800656, 0.71332634715229048141580800656, 1.88266571798756806422390956403, 2.72325357697640913470240784477, 3.97626141773580400613243344731, 4.93267325054346283417616782638, 5.91094702021289906999846783998, 6.68523229061371586426743717277, 7.30638868497843428313417528525, 8.790238837846345152569362530670, 9.136405585163481890600679569711

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