Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 0.305·5-s − 0.140·7-s + 9-s + 1.85·11-s − 1.94·13-s + 0.305·15-s + 5.66·17-s − 1.04·19-s + 0.140·21-s + 23-s − 4.90·25-s − 27-s − 29-s − 6.72·31-s − 1.85·33-s + 0.0428·35-s − 1.42·37-s + 1.94·39-s + 0.522·41-s + 4.27·43-s − 0.305·45-s + 7.07·47-s − 6.98·49-s − 5.66·51-s − 11.0·53-s − 0.568·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.136·5-s − 0.0529·7-s + 0.333·9-s + 0.560·11-s − 0.538·13-s + 0.0789·15-s + 1.37·17-s − 0.240·19-s + 0.0305·21-s + 0.208·23-s − 0.981·25-s − 0.192·27-s − 0.185·29-s − 1.20·31-s − 0.323·33-s + 0.00723·35-s − 0.234·37-s + 0.311·39-s + 0.0816·41-s + 0.652·43-s − 0.0455·45-s + 1.03·47-s − 0.997·49-s − 0.793·51-s − 1.51·53-s − 0.0766·55-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 \( 1 + 0.305T + 5T^{2} \)
7 \( 1 + 0.140T + 7T^{2} \)
11 \( 1 - 1.85T + 11T^{2} \)
13 \( 1 + 1.94T + 13T^{2} \)
17 \( 1 - 5.66T + 17T^{2} \)
19 \( 1 + 1.04T + 19T^{2} \)
31 \( 1 + 6.72T + 31T^{2} \)
37 \( 1 + 1.42T + 37T^{2} \)
41 \( 1 - 0.522T + 41T^{2} \)
43 \( 1 - 4.27T + 43T^{2} \)
47 \( 1 - 7.07T + 47T^{2} \)
53 \( 1 + 11.0T + 53T^{2} \)
59 \( 1 + 10.0T + 59T^{2} \)
61 \( 1 - 7.67T + 61T^{2} \)
67 \( 1 + 12.0T + 67T^{2} \)
71 \( 1 - 8.29T + 71T^{2} \)
73 \( 1 + 0.0686T + 73T^{2} \)
79 \( 1 - 8.58T + 79T^{2} \)
83 \( 1 - 7.65T + 83T^{2} \)
89 \( 1 - 8.93T + 89T^{2} \)
97 \( 1 - 4.43T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.63832302604901244375852517683, −6.74438561877811091472797996440, −6.06390539679480133066993507395, −5.44862837420902125625548307256, −4.75132834337730233409644747293, −3.88755068121190086244519408659, −3.26630463919963835991186350133, −2.10309108356817580903630521657, −1.19367625317729728681209256683, 0, 1.19367625317729728681209256683, 2.10309108356817580903630521657, 3.26630463919963835991186350133, 3.88755068121190086244519408659, 4.75132834337730233409644747293, 5.44862837420902125625548307256, 6.06390539679480133066993507395, 6.74438561877811091472797996440, 7.63832302604901244375852517683

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