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 + 2.64·5-s − 1.93·7-s + 9-s − 1.24·11-s − 1.48·13-s − 2.64·15-s − 4.24·17-s + 2.85·19-s + 1.93·21-s + 23-s + 2.02·25-s − 27-s − 29-s + 8.48·31-s + 1.24·33-s − 5.12·35-s + 0.788·37-s + 1.48·39-s − 12.2·41-s + 11.0·43-s + 2.64·45-s − 4.51·47-s − 3.25·49-s + 4.24·51-s + 10.2·53-s − 3.28·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.18·5-s − 0.730·7-s + 0.333·9-s − 0.374·11-s − 0.412·13-s − 0.684·15-s − 1.02·17-s + 0.655·19-s + 0.422·21-s + 0.208·23-s + 0.404·25-s − 0.192·27-s − 0.185·29-s + 1.52·31-s + 0.216·33-s − 0.866·35-s + 0.129·37-s + 0.238·39-s − 1.91·41-s + 1.67·43-s + 0.395·45-s − 0.658·47-s − 0.465·49-s + 0.593·51-s + 1.40·53-s − 0.443·55-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 \( 1 - 2.64T + 5T^{2} \)
7 \( 1 + 1.93T + 7T^{2} \)
11 \( 1 + 1.24T + 11T^{2} \)
13 \( 1 + 1.48T + 13T^{2} \)
17 \( 1 + 4.24T + 17T^{2} \)
19 \( 1 - 2.85T + 19T^{2} \)
31 \( 1 - 8.48T + 31T^{2} \)
37 \( 1 - 0.788T + 37T^{2} \)
41 \( 1 + 12.2T + 41T^{2} \)
43 \( 1 - 11.0T + 43T^{2} \)
47 \( 1 + 4.51T + 47T^{2} \)
53 \( 1 - 10.2T + 53T^{2} \)
59 \( 1 + 1.07T + 59T^{2} \)
61 \( 1 + 1.98T + 61T^{2} \)
67 \( 1 - 0.801T + 67T^{2} \)
71 \( 1 - 1.24T + 71T^{2} \)
73 \( 1 - 1.47T + 73T^{2} \)
79 \( 1 + 12.7T + 79T^{2} \)
83 \( 1 - 3.32T + 83T^{2} \)
89 \( 1 - 2.02T + 89T^{2} \)
97 \( 1 - 9.39T + 97T^{2} \)
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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

−7.29152457195895640290528567913, −6.58476991725406175412244806101, −6.20980310599026240483471123488, −5.40183777598493592276156874533, −4.90887445921022930831079014952, −3.98469016922687449838940250076, −2.90148985555692839501532989154, −2.28391241108589756952100231907, −1.24212863037046833228621667519, 0, 1.24212863037046833228621667519, 2.28391241108589756952100231907, 2.90148985555692839501532989154, 3.98469016922687449838940250076, 4.90887445921022930831079014952, 5.40183777598493592276156874533, 6.20980310599026240483471123488, 6.58476991725406175412244806101, 7.29152457195895640290528567913

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