Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 167 $
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 − 1.85·5-s + 2.41·7-s + 9-s − 2.49·11-s + 0.116·13-s − 1.85·15-s − 6.12·17-s − 3.34·19-s + 2.41·21-s − 8.19·23-s − 1.57·25-s + 27-s + 0.0936·29-s + 1.44·31-s − 2.49·33-s − 4.47·35-s − 0.230·37-s + 0.116·39-s − 4.49·41-s + 5.35·43-s − 1.85·45-s + 7.01·47-s − 1.16·49-s − 6.12·51-s + 3.67·53-s + 4.62·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.828·5-s + 0.913·7-s + 0.333·9-s − 0.752·11-s + 0.0322·13-s − 0.478·15-s − 1.48·17-s − 0.766·19-s + 0.527·21-s − 1.70·23-s − 0.314·25-s + 0.192·27-s + 0.0173·29-s + 0.260·31-s − 0.434·33-s − 0.756·35-s − 0.0378·37-s + 0.0186·39-s − 0.702·41-s + 0.816·43-s − 0.276·45-s + 1.02·47-s − 0.166·49-s − 0.858·51-s + 0.505·53-s + 0.623·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(2004\)    =    \(2^{2} \cdot 3 \cdot 167\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{2004} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 2004,\ (\ :1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;167\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;167\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
167 \( 1 + T \)
good5 \( 1 + 1.85T + 5T^{2} \)
7 \( 1 - 2.41T + 7T^{2} \)
11 \( 1 + 2.49T + 11T^{2} \)
13 \( 1 - 0.116T + 13T^{2} \)
17 \( 1 + 6.12T + 17T^{2} \)
19 \( 1 + 3.34T + 19T^{2} \)
23 \( 1 + 8.19T + 23T^{2} \)
29 \( 1 - 0.0936T + 29T^{2} \)
31 \( 1 - 1.44T + 31T^{2} \)
37 \( 1 + 0.230T + 37T^{2} \)
41 \( 1 + 4.49T + 41T^{2} \)
43 \( 1 - 5.35T + 43T^{2} \)
47 \( 1 - 7.01T + 47T^{2} \)
53 \( 1 - 3.67T + 53T^{2} \)
59 \( 1 + 1.76T + 59T^{2} \)
61 \( 1 - 8.66T + 61T^{2} \)
67 \( 1 + 15.1T + 67T^{2} \)
71 \( 1 + 8.51T + 71T^{2} \)
73 \( 1 - 4.74T + 73T^{2} \)
79 \( 1 + 3.24T + 79T^{2} \)
83 \( 1 + 0.437T + 83T^{2} \)
89 \( 1 - 11.9T + 89T^{2} \)
97 \( 1 - 5.72T + 97T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.490679673889379652380962576248, −8.109070790878317608417756232491, −7.45865595534767831120342324390, −6.52567157981902916187048400370, −5.46909155274823512957530644414, −4.34838078365187692253764561309, −4.05279710399523654098746584766, −2.65136481485911717227223386597, −1.83992559203755456349688191873, 0, 1.83992559203755456349688191873, 2.65136481485911717227223386597, 4.05279710399523654098746584766, 4.34838078365187692253764561309, 5.46909155274823512957530644414, 6.52567157981902916187048400370, 7.45865595534767831120342324390, 8.109070790878317608417756232491, 8.490679673889379652380962576248

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