Properties

Label 2-8004-1.1-c1-0-40
Degree $2$
Conductor $8004$
Sign $-1$
Analytic cond. $63.9122$
Root an. cond. $7.99451$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 4.15·5-s − 1.11·7-s + 9-s − 5.32·11-s + 0.315·13-s + 4.15·15-s − 1.36·17-s + 5.63·19-s + 1.11·21-s + 23-s + 12.2·25-s − 27-s − 29-s − 7.05·31-s + 5.32·33-s + 4.63·35-s − 5.05·37-s − 0.315·39-s + 2.34·41-s + 4.48·43-s − 4.15·45-s + 3.25·47-s − 5.75·49-s + 1.36·51-s + 7.64·53-s + 22.1·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.85·5-s − 0.421·7-s + 0.333·9-s − 1.60·11-s + 0.0875·13-s + 1.07·15-s − 0.331·17-s + 1.29·19-s + 0.243·21-s + 0.208·23-s + 2.45·25-s − 0.192·27-s − 0.185·29-s − 1.26·31-s + 0.926·33-s + 0.782·35-s − 0.831·37-s − 0.0505·39-s + 0.366·41-s + 0.683·43-s − 0.619·45-s + 0.474·47-s − 0.822·49-s + 0.191·51-s + 1.05·53-s + 2.98·55-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: $-1$
Analytic conductor: \(63.9122\)
Root analytic conductor: \(7.99451\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
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 \( 1 + 4.15T + 5T^{2} \)
7 \( 1 + 1.11T + 7T^{2} \)
11 \( 1 + 5.32T + 11T^{2} \)
13 \( 1 - 0.315T + 13T^{2} \)
17 \( 1 + 1.36T + 17T^{2} \)
19 \( 1 - 5.63T + 19T^{2} \)
31 \( 1 + 7.05T + 31T^{2} \)
37 \( 1 + 5.05T + 37T^{2} \)
41 \( 1 - 2.34T + 41T^{2} \)
43 \( 1 - 4.48T + 43T^{2} \)
47 \( 1 - 3.25T + 47T^{2} \)
53 \( 1 - 7.64T + 53T^{2} \)
59 \( 1 + 5.11T + 59T^{2} \)
61 \( 1 - 1.44T + 61T^{2} \)
67 \( 1 - 1.32T + 67T^{2} \)
71 \( 1 - 5.18T + 71T^{2} \)
73 \( 1 - 2.17T + 73T^{2} \)
79 \( 1 - 16.7T + 79T^{2} \)
83 \( 1 - 10.5T + 83T^{2} \)
89 \( 1 - 7.27T + 89T^{2} \)
97 \( 1 - 17.8T + 97T^{2} \)
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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

−7.50822380097024663176854950533, −7.06775866393348485055934950011, −6.09704104679721474706512571789, −5.15162039295950633219762665301, −4.85925537850983143515617952089, −3.74665097988547237566769199685, −3.37804367751554691220155940984, −2.37223667776865014923104019748, −0.807268740114146943614618514108, 0, 0.807268740114146943614618514108, 2.37223667776865014923104019748, 3.37804367751554691220155940984, 3.74665097988547237566769199685, 4.85925537850983143515617952089, 5.15162039295950633219762665301, 6.09704104679721474706512571789, 7.06775866393348485055934950011, 7.50822380097024663176854950533

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