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 − 1.38·5-s − 0.871·7-s + 9-s − 5.74·11-s + 3.90·13-s − 1.38·15-s − 0.477·17-s + 2.16·19-s − 0.871·21-s + 3.53·23-s − 3.07·25-s + 27-s − 5.05·29-s + 3.41·31-s − 5.74·33-s + 1.21·35-s − 4.98·37-s + 3.90·39-s − 5.50·41-s − 3.83·43-s − 1.38·45-s − 13.6·47-s − 6.23·49-s − 0.477·51-s − 11.6·53-s + 7.98·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.621·5-s − 0.329·7-s + 0.333·9-s − 1.73·11-s + 1.08·13-s − 0.358·15-s − 0.115·17-s + 0.495·19-s − 0.190·21-s + 0.736·23-s − 0.614·25-s + 0.192·27-s − 0.938·29-s + 0.613·31-s − 1.00·33-s + 0.204·35-s − 0.820·37-s + 0.625·39-s − 0.860·41-s − 0.584·43-s − 0.207·45-s − 1.99·47-s − 0.891·49-s − 0.0669·51-s − 1.60·53-s + 1.07·55-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 \( 1 + 1.38T + 5T^{2} \)
7 \( 1 + 0.871T + 7T^{2} \)
11 \( 1 + 5.74T + 11T^{2} \)
13 \( 1 - 3.90T + 13T^{2} \)
17 \( 1 + 0.477T + 17T^{2} \)
19 \( 1 - 2.16T + 19T^{2} \)
23 \( 1 - 3.53T + 23T^{2} \)
29 \( 1 + 5.05T + 29T^{2} \)
31 \( 1 - 3.41T + 31T^{2} \)
37 \( 1 + 4.98T + 37T^{2} \)
41 \( 1 + 5.50T + 41T^{2} \)
43 \( 1 + 3.83T + 43T^{2} \)
47 \( 1 + 13.6T + 47T^{2} \)
53 \( 1 + 11.6T + 53T^{2} \)
59 \( 1 + 0.528T + 59T^{2} \)
61 \( 1 - 1.27T + 61T^{2} \)
67 \( 1 - 8.13T + 67T^{2} \)
71 \( 1 - 4.17T + 71T^{2} \)
73 \( 1 + 6.12T + 73T^{2} \)
79 \( 1 + 15.4T + 79T^{2} \)
83 \( 1 + 1.01T + 83T^{2} \)
89 \( 1 + 9.40T + 89T^{2} \)
97 \( 1 + 0.321T + 97T^{2} \)
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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

−8.502093327911818181402676429314, −8.119185404356004154002279288876, −7.38026537193751277003205469189, −6.51190476727601604255819513059, −5.46428084943718152382330818906, −4.68048858821666511717350231746, −3.49467658768158127292921290469, −3.03358858914212224293941068619, −1.71105587019226403451880842145, 0, 1.71105587019226403451880842145, 3.03358858914212224293941068619, 3.49467658768158127292921290469, 4.68048858821666511717350231746, 5.46428084943718152382330818906, 6.51190476727601604255819513059, 7.38026537193751277003205469189, 8.119185404356004154002279288876, 8.502093327911818181402676429314

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