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 + 0.0891·5-s + 2.52·7-s + 9-s − 1.55·11-s + 6.31·13-s − 0.0891·15-s − 6.02·17-s + 4.11·19-s − 2.52·21-s + 23-s − 4.99·25-s − 27-s − 29-s − 6.28·31-s + 1.55·33-s + 0.224·35-s − 3.14·37-s − 6.31·39-s − 3.88·41-s + 1.91·43-s + 0.0891·45-s − 12.5·47-s − 0.645·49-s + 6.02·51-s − 3.50·53-s − 0.138·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.0398·5-s + 0.952·7-s + 0.333·9-s − 0.467·11-s + 1.75·13-s − 0.0230·15-s − 1.46·17-s + 0.945·19-s − 0.550·21-s + 0.208·23-s − 0.998·25-s − 0.192·27-s − 0.185·29-s − 1.12·31-s + 0.270·33-s + 0.0379·35-s − 0.516·37-s − 1.01·39-s − 0.606·41-s + 0.292·43-s + 0.0132·45-s − 1.82·47-s − 0.0921·49-s + 0.843·51-s − 0.481·53-s − 0.0186·55-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 \( 1 - 0.0891T + 5T^{2} \)
7 \( 1 - 2.52T + 7T^{2} \)
11 \( 1 + 1.55T + 11T^{2} \)
13 \( 1 - 6.31T + 13T^{2} \)
17 \( 1 + 6.02T + 17T^{2} \)
19 \( 1 - 4.11T + 19T^{2} \)
31 \( 1 + 6.28T + 31T^{2} \)
37 \( 1 + 3.14T + 37T^{2} \)
41 \( 1 + 3.88T + 41T^{2} \)
43 \( 1 - 1.91T + 43T^{2} \)
47 \( 1 + 12.5T + 47T^{2} \)
53 \( 1 + 3.50T + 53T^{2} \)
59 \( 1 + 9.16T + 59T^{2} \)
61 \( 1 - 10.3T + 61T^{2} \)
67 \( 1 + 1.85T + 67T^{2} \)
71 \( 1 - 2.06T + 71T^{2} \)
73 \( 1 + 0.852T + 73T^{2} \)
79 \( 1 + 14.6T + 79T^{2} \)
83 \( 1 - 1.87T + 83T^{2} \)
89 \( 1 + 11.8T + 89T^{2} \)
97 \( 1 + 5.06T + 97T^{2} \)
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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

−7.48776369901954375924835746377, −6.74207295403015515157052303689, −6.04568748627271719976862442418, −5.40341377788925110690124710066, −4.76757011670533176639862710531, −3.98777231361848310031508296191, −3.22509120441562922238909594297, −1.94039604880786427856304263683, −1.36880932160754387390361150958, 0, 1.36880932160754387390361150958, 1.94039604880786427856304263683, 3.22509120441562922238909594297, 3.98777231361848310031508296191, 4.76757011670533176639862710531, 5.40341377788925110690124710066, 6.04568748627271719976862442418, 6.74207295403015515157052303689, 7.48776369901954375924835746377

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