Properties

Degree $2$
Conductor $8004$
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 + 1.27·5-s + 1.05·7-s + 9-s + 2.86·11-s − 0.442·13-s − 1.27·15-s + 7.11·17-s + 5.55·19-s − 1.05·21-s + 23-s − 3.37·25-s − 27-s + 29-s + 9.86·31-s − 2.86·33-s + 1.35·35-s + 3.13·37-s + 0.442·39-s + 0.570·41-s + 3.31·43-s + 1.27·45-s + 7.62·47-s − 5.87·49-s − 7.11·51-s − 0.0892·53-s + 3.65·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.570·5-s + 0.400·7-s + 0.333·9-s + 0.865·11-s − 0.122·13-s − 0.329·15-s + 1.72·17-s + 1.27·19-s − 0.231·21-s + 0.208·23-s − 0.674·25-s − 0.192·27-s + 0.185·29-s + 1.77·31-s − 0.499·33-s + 0.228·35-s + 0.515·37-s + 0.0708·39-s + 0.0890·41-s + 0.505·43-s + 0.190·45-s + 1.11·47-s − 0.839·49-s − 0.996·51-s − 0.0122·53-s + 0.493·55-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: $1$
Motivic weight: \(1\)
Character: $\chi_{8004} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8004,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.572235990\)
\(L(\frac12)\) \(\approx\) \(2.572235990\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.27T + 5T^{2} \)
7 \( 1 - 1.05T + 7T^{2} \)
11 \( 1 - 2.86T + 11T^{2} \)
13 \( 1 + 0.442T + 13T^{2} \)
17 \( 1 - 7.11T + 17T^{2} \)
19 \( 1 - 5.55T + 19T^{2} \)
31 \( 1 - 9.86T + 31T^{2} \)
37 \( 1 - 3.13T + 37T^{2} \)
41 \( 1 - 0.570T + 41T^{2} \)
43 \( 1 - 3.31T + 43T^{2} \)
47 \( 1 - 7.62T + 47T^{2} \)
53 \( 1 + 0.0892T + 53T^{2} \)
59 \( 1 + 7.70T + 59T^{2} \)
61 \( 1 - 0.605T + 61T^{2} \)
67 \( 1 + 2.08T + 67T^{2} \)
71 \( 1 + 9.62T + 71T^{2} \)
73 \( 1 - 9.39T + 73T^{2} \)
79 \( 1 + 4.40T + 79T^{2} \)
83 \( 1 + 11.8T + 83T^{2} \)
89 \( 1 + 15.9T + 89T^{2} \)
97 \( 1 - 1.77T + 97T^{2} \)
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

−7.66575417343926053304881192488, −7.23912021445213070125538196466, −6.18577195409111424422141536746, −5.88405520912854984203443763590, −5.10757132427326860000740058318, −4.43598518350197037237944794318, −3.51444220128092206614781670333, −2.69322278893921265196700949623, −1.47389304756662621446749955226, −0.937894510332341865043757388756, 0.937894510332341865043757388756, 1.47389304756662621446749955226, 2.69322278893921265196700949623, 3.51444220128092206614781670333, 4.43598518350197037237944794318, 5.10757132427326860000740058318, 5.88405520912854984203443763590, 6.18577195409111424422141536746, 7.23912021445213070125538196466, 7.66575417343926053304881192488

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