Properties

Degree $2$
Conductor $8004$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 0.650·5-s − 1.33·7-s + 9-s + 5.13·11-s + 1.17·13-s + 0.650·15-s − 0.132·17-s − 1.33·19-s + 1.33·21-s − 23-s − 4.57·25-s − 27-s + 29-s − 1.92·31-s − 5.13·33-s + 0.868·35-s − 2.05·37-s − 1.17·39-s − 10.6·41-s + 5.90·43-s − 0.650·45-s − 6.07·47-s − 5.22·49-s + 0.132·51-s + 13.1·53-s − 3.34·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.291·5-s − 0.504·7-s + 0.333·9-s + 1.54·11-s + 0.326·13-s + 0.168·15-s − 0.0321·17-s − 0.305·19-s + 0.291·21-s − 0.208·23-s − 0.915·25-s − 0.192·27-s + 0.185·29-s − 0.346·31-s − 0.893·33-s + 0.146·35-s − 0.338·37-s − 0.188·39-s − 1.66·41-s + 0.900·43-s − 0.0970·45-s − 0.886·47-s − 0.745·49-s + 0.0185·51-s + 1.80·53-s − 0.450·55-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: \(1\)
Selberg data: \((2,\ 8004,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.650T + 5T^{2} \)
7 \( 1 + 1.33T + 7T^{2} \)
11 \( 1 - 5.13T + 11T^{2} \)
13 \( 1 - 1.17T + 13T^{2} \)
17 \( 1 + 0.132T + 17T^{2} \)
19 \( 1 + 1.33T + 19T^{2} \)
31 \( 1 + 1.92T + 31T^{2} \)
37 \( 1 + 2.05T + 37T^{2} \)
41 \( 1 + 10.6T + 41T^{2} \)
43 \( 1 - 5.90T + 43T^{2} \)
47 \( 1 + 6.07T + 47T^{2} \)
53 \( 1 - 13.1T + 53T^{2} \)
59 \( 1 + 5.88T + 59T^{2} \)
61 \( 1 - 9.63T + 61T^{2} \)
67 \( 1 + 5.90T + 67T^{2} \)
71 \( 1 + 7.38T + 71T^{2} \)
73 \( 1 - 4.38T + 73T^{2} \)
79 \( 1 + 7.87T + 79T^{2} \)
83 \( 1 - 17.7T + 83T^{2} \)
89 \( 1 - 9.08T + 89T^{2} \)
97 \( 1 + 16.7T + 97T^{2} \)
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

−7.30726240825801532198660364372, −6.68968568865006803572508636774, −6.21967766821280868934317063736, −5.51608026124500734195142322662, −4.58720756440243804415801952088, −3.87020403415182852603176353015, −3.38247841490585550342328880751, −2.06991288056023205278420756403, −1.18024230486770180626745678985, 0, 1.18024230486770180626745678985, 2.06991288056023205278420756403, 3.38247841490585550342328880751, 3.87020403415182852603176353015, 4.58720756440243804415801952088, 5.51608026124500734195142322662, 6.21967766821280868934317063736, 6.68968568865006803572508636774, 7.30726240825801532198660364372

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