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 + 0.900·5-s + 3.90·7-s + 9-s − 5.39·11-s − 2.18·13-s − 0.900·15-s − 2.64·17-s + 3.37·19-s − 3.90·21-s − 23-s − 4.18·25-s − 27-s + 29-s + 5.08·31-s + 5.39·33-s + 3.51·35-s − 3.79·37-s + 2.18·39-s + 8.82·41-s + 0.613·43-s + 0.900·45-s + 1.02·47-s + 8.24·49-s + 2.64·51-s + 5.94·53-s − 4.86·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.402·5-s + 1.47·7-s + 0.333·9-s − 1.62·11-s − 0.606·13-s − 0.232·15-s − 0.641·17-s + 0.773·19-s − 0.852·21-s − 0.208·23-s − 0.837·25-s − 0.192·27-s + 0.185·29-s + 0.913·31-s + 0.939·33-s + 0.594·35-s − 0.623·37-s + 0.349·39-s + 1.37·41-s + 0.0935·43-s + 0.134·45-s + 0.149·47-s + 1.17·49-s + 0.370·51-s + 0.816·53-s − 0.655·55-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 \( 1 - 0.900T + 5T^{2} \)
7 \( 1 - 3.90T + 7T^{2} \)
11 \( 1 + 5.39T + 11T^{2} \)
13 \( 1 + 2.18T + 13T^{2} \)
17 \( 1 + 2.64T + 17T^{2} \)
19 \( 1 - 3.37T + 19T^{2} \)
31 \( 1 - 5.08T + 31T^{2} \)
37 \( 1 + 3.79T + 37T^{2} \)
41 \( 1 - 8.82T + 41T^{2} \)
43 \( 1 - 0.613T + 43T^{2} \)
47 \( 1 - 1.02T + 47T^{2} \)
53 \( 1 - 5.94T + 53T^{2} \)
59 \( 1 + 10.0T + 59T^{2} \)
61 \( 1 + 3.09T + 61T^{2} \)
67 \( 1 + 12.6T + 67T^{2} \)
71 \( 1 + 4.69T + 71T^{2} \)
73 \( 1 - 9.03T + 73T^{2} \)
79 \( 1 + 3.30T + 79T^{2} \)
83 \( 1 + 1.01T + 83T^{2} \)
89 \( 1 + 17.4T + 89T^{2} \)
97 \( 1 - 4.91T + 97T^{2} \)
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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

−7.68728753430991665362881938926, −6.87309612414701422486535133353, −5.81057178030045534614511799534, −5.44628066567346403393969939634, −4.74430304205988557608171391378, −4.26746481367675191393026223490, −2.85326570971119426892813440256, −2.20027547337030882009090717352, −1.29771476150099003287640726054, 0, 1.29771476150099003287640726054, 2.20027547337030882009090717352, 2.85326570971119426892813440256, 4.26746481367675191393026223490, 4.74430304205988557608171391378, 5.44628066567346403393969939634, 5.81057178030045534614511799534, 6.87309612414701422486535133353, 7.68728753430991665362881938926

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