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 + 2.30·5-s − 4.14·7-s + 9-s + 1.07·11-s + 7.08·13-s − 2.30·15-s + 6.59·17-s − 6.26·19-s + 4.14·21-s + 23-s + 0.294·25-s − 27-s + 29-s − 5.73·31-s − 1.07·33-s − 9.54·35-s + 9.95·37-s − 7.08·39-s − 9.49·41-s + 10.2·43-s + 2.30·45-s + 6.32·47-s + 10.1·49-s − 6.59·51-s − 11.2·53-s + 2.47·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.02·5-s − 1.56·7-s + 0.333·9-s + 0.324·11-s + 1.96·13-s − 0.594·15-s + 1.59·17-s − 1.43·19-s + 0.904·21-s + 0.208·23-s + 0.0588·25-s − 0.192·27-s + 0.185·29-s − 1.02·31-s − 0.187·33-s − 1.61·35-s + 1.63·37-s − 1.13·39-s − 1.48·41-s + 1.56·43-s + 0.343·45-s + 0.922·47-s + 1.45·49-s − 0.923·51-s − 1.54·53-s + 0.333·55-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

\(L(1)\) \(\approx\) \(1.903670167\)
\(L(\frac12)\) \(\approx\) \(1.903670167\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.30T + 5T^{2} \)
7 \( 1 + 4.14T + 7T^{2} \)
11 \( 1 - 1.07T + 11T^{2} \)
13 \( 1 - 7.08T + 13T^{2} \)
17 \( 1 - 6.59T + 17T^{2} \)
19 \( 1 + 6.26T + 19T^{2} \)
31 \( 1 + 5.73T + 31T^{2} \)
37 \( 1 - 9.95T + 37T^{2} \)
41 \( 1 + 9.49T + 41T^{2} \)
43 \( 1 - 10.2T + 43T^{2} \)
47 \( 1 - 6.32T + 47T^{2} \)
53 \( 1 + 11.2T + 53T^{2} \)
59 \( 1 - 12.4T + 59T^{2} \)
61 \( 1 - 0.914T + 61T^{2} \)
67 \( 1 - 0.117T + 67T^{2} \)
71 \( 1 - 5.86T + 71T^{2} \)
73 \( 1 - 11.6T + 73T^{2} \)
79 \( 1 + 9.13T + 79T^{2} \)
83 \( 1 + 2.02T + 83T^{2} \)
89 \( 1 + 5.19T + 89T^{2} \)
97 \( 1 - 11.6T + 97T^{2} \)
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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

−7.81227734026394672127306732293, −6.75012292245967752452482674468, −6.36103739620430653241892387517, −5.86731711384540822645096207881, −5.42868939408629136735942126845, −4.06594120443781171195907229673, −3.62931265302688637579204351888, −2.71131559910019141770111963589, −1.62003339473561120201652171924, −0.73560704391702737636059225823, 0.73560704391702737636059225823, 1.62003339473561120201652171924, 2.71131559910019141770111963589, 3.62931265302688637579204351888, 4.06594120443781171195907229673, 5.42868939408629136735942126845, 5.86731711384540822645096207881, 6.36103739620430653241892387517, 6.75012292245967752452482674468, 7.81227734026394672127306732293

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