Properties

Degree $2$
Conductor $8004$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 3.33·5-s − 2.20·7-s + 9-s + 4.06·11-s + 6.44·13-s + 3.33·15-s + 6.35·17-s + 6.75·19-s + 2.20·21-s + 23-s + 6.15·25-s − 27-s + 29-s + 3.50·31-s − 4.06·33-s + 7.37·35-s + 1.43·37-s − 6.44·39-s + 10.9·41-s − 11.9·43-s − 3.33·45-s − 5.99·47-s − 2.11·49-s − 6.35·51-s − 4.14·53-s − 13.5·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.49·5-s − 0.835·7-s + 0.333·9-s + 1.22·11-s + 1.78·13-s + 0.862·15-s + 1.54·17-s + 1.54·19-s + 0.482·21-s + 0.208·23-s + 1.23·25-s − 0.192·27-s + 0.185·29-s + 0.630·31-s − 0.707·33-s + 1.24·35-s + 0.235·37-s − 1.03·39-s + 1.71·41-s − 1.81·43-s − 0.497·45-s − 0.874·47-s − 0.302·49-s − 0.889·51-s − 0.569·53-s − 1.83·55-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

\(L(1)\) \(\approx\) \(1.551411512\)
\(L(\frac12)\) \(\approx\) \(1.551411512\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.33T + 5T^{2} \)
7 \( 1 + 2.20T + 7T^{2} \)
11 \( 1 - 4.06T + 11T^{2} \)
13 \( 1 - 6.44T + 13T^{2} \)
17 \( 1 - 6.35T + 17T^{2} \)
19 \( 1 - 6.75T + 19T^{2} \)
31 \( 1 - 3.50T + 31T^{2} \)
37 \( 1 - 1.43T + 37T^{2} \)
41 \( 1 - 10.9T + 41T^{2} \)
43 \( 1 + 11.9T + 43T^{2} \)
47 \( 1 + 5.99T + 47T^{2} \)
53 \( 1 + 4.14T + 53T^{2} \)
59 \( 1 - 1.75T + 59T^{2} \)
61 \( 1 - 15.3T + 61T^{2} \)
67 \( 1 + 2.00T + 67T^{2} \)
71 \( 1 - 16.1T + 71T^{2} \)
73 \( 1 + 4.96T + 73T^{2} \)
79 \( 1 + 6.38T + 79T^{2} \)
83 \( 1 - 13.0T + 83T^{2} \)
89 \( 1 - 6.82T + 89T^{2} \)
97 \( 1 - 3.18T + 97T^{2} \)
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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

−7.957919483398108284659180539683, −7.00795787825232140639775782294, −6.53589662456220126886256118283, −5.85336110860989664487703601515, −5.05031073490407217617408260892, −4.05255856118469376769820223978, −3.53337350632378209281943114981, −3.18174457799133408198328838991, −1.26614206233050415912771756557, −0.76975637896183821871004633539, 0.76975637896183821871004633539, 1.26614206233050415912771756557, 3.18174457799133408198328838991, 3.53337350632378209281943114981, 4.05255856118469376769820223978, 5.05031073490407217617408260892, 5.85336110860989664487703601515, 6.53589662456220126886256118283, 7.00795787825232140639775782294, 7.957919483398108284659180539683

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