Properties

Label 2-8004-1.1-c1-0-38
Degree $2$
Conductor $8004$
Sign $1$
Analytic cond. $63.9122$
Root an. cond. $7.99451$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 2.96·5-s − 3.61·7-s + 9-s − 1.51·11-s + 0.595·13-s + 2.96·15-s + 0.748·17-s − 3.13·19-s − 3.61·21-s + 23-s + 3.81·25-s + 27-s − 29-s + 1.50·31-s − 1.51·33-s − 10.7·35-s + 3.65·37-s + 0.595·39-s − 0.901·41-s − 7.23·43-s + 2.96·45-s + 6.53·47-s + 6.07·49-s + 0.748·51-s + 11.1·53-s − 4.50·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.32·5-s − 1.36·7-s + 0.333·9-s − 0.457·11-s + 0.165·13-s + 0.766·15-s + 0.181·17-s − 0.719·19-s − 0.788·21-s + 0.208·23-s + 0.763·25-s + 0.192·27-s − 0.185·29-s + 0.270·31-s − 0.264·33-s − 1.81·35-s + 0.601·37-s + 0.0954·39-s − 0.140·41-s − 1.10·43-s + 0.442·45-s + 0.953·47-s + 0.867·49-s + 0.104·51-s + 1.52·53-s − 0.607·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8004\)    =    \(2^{2} \cdot 3 \cdot 23 \cdot 29\)
Sign: $1$
Analytic conductor: \(63.9122\)
Root analytic conductor: \(7.99451\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8004,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.699369504\)
\(L(\frac12)\) \(\approx\) \(2.699369504\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
23 \( 1 - T \)
29 \( 1 + T \)
good5 \( 1 - 2.96T + 5T^{2} \)
7 \( 1 + 3.61T + 7T^{2} \)
11 \( 1 + 1.51T + 11T^{2} \)
13 \( 1 - 0.595T + 13T^{2} \)
17 \( 1 - 0.748T + 17T^{2} \)
19 \( 1 + 3.13T + 19T^{2} \)
31 \( 1 - 1.50T + 31T^{2} \)
37 \( 1 - 3.65T + 37T^{2} \)
41 \( 1 + 0.901T + 41T^{2} \)
43 \( 1 + 7.23T + 43T^{2} \)
47 \( 1 - 6.53T + 47T^{2} \)
53 \( 1 - 11.1T + 53T^{2} \)
59 \( 1 - 14.9T + 59T^{2} \)
61 \( 1 - 4.00T + 61T^{2} \)
67 \( 1 - 7.19T + 67T^{2} \)
71 \( 1 - 7.23T + 71T^{2} \)
73 \( 1 - 8.91T + 73T^{2} \)
79 \( 1 - 8.63T + 79T^{2} \)
83 \( 1 + 7.09T + 83T^{2} \)
89 \( 1 - 14.5T + 89T^{2} \)
97 \( 1 + 7.05T + 97T^{2} \)
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

−7.940729565740858975984604709144, −6.80647526351939962056211456274, −6.65656729723729512419737906733, −5.73693678128754861870327811149, −5.27102979838816971135550929935, −4.11847598261819518822248610575, −3.39824165135558593801879354240, −2.54609175064899504735305159912, −2.07149862211835433337128095158, −0.77111572114126963912441107576, 0.77111572114126963912441107576, 2.07149862211835433337128095158, 2.54609175064899504735305159912, 3.39824165135558593801879354240, 4.11847598261819518822248610575, 5.27102979838816971135550929935, 5.73693678128754861870327811149, 6.65656729723729512419737906733, 6.80647526351939962056211456274, 7.940729565740858975984604709144

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