Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 23 \cdot 29 $
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 − 2.77·5-s + 0.556·7-s + 9-s + 3.82·11-s − 0.0549·13-s − 2.77·15-s − 2.00·17-s − 2.76·19-s + 0.556·21-s + 23-s + 2.69·25-s + 27-s + 29-s − 3.02·31-s + 3.82·33-s − 1.54·35-s + 1.22·37-s − 0.0549·39-s − 6.65·41-s − 4.40·43-s − 2.77·45-s − 4.00·47-s − 6.69·49-s − 2.00·51-s − 3.40·53-s − 10.6·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.24·5-s + 0.210·7-s + 0.333·9-s + 1.15·11-s − 0.0152·13-s − 0.716·15-s − 0.486·17-s − 0.634·19-s + 0.121·21-s + 0.208·23-s + 0.539·25-s + 0.192·27-s + 0.185·29-s − 0.544·31-s + 0.666·33-s − 0.260·35-s + 0.200·37-s − 0.00880·39-s − 1.03·41-s − 0.671·43-s − 0.413·45-s − 0.584·47-s − 0.955·49-s − 0.281·51-s − 0.467·53-s − 1.43·55-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

\( d \)  =  \(2\)
\( N \)  =  \(8004\)    =    \(2^{2} \cdot 3 \cdot 23 \cdot 29\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{8004} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 8004,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;23,\;29\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;23,\;29\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
3 \( 1 - T \)
23 \( 1 - T \)
29 \( 1 - T \)
good5 \( 1 + 2.77T + 5T^{2} \)
7 \( 1 - 0.556T + 7T^{2} \)
11 \( 1 - 3.82T + 11T^{2} \)
13 \( 1 + 0.0549T + 13T^{2} \)
17 \( 1 + 2.00T + 17T^{2} \)
19 \( 1 + 2.76T + 19T^{2} \)
31 \( 1 + 3.02T + 31T^{2} \)
37 \( 1 - 1.22T + 37T^{2} \)
41 \( 1 + 6.65T + 41T^{2} \)
43 \( 1 + 4.40T + 43T^{2} \)
47 \( 1 + 4.00T + 47T^{2} \)
53 \( 1 + 3.40T + 53T^{2} \)
59 \( 1 - 10.2T + 59T^{2} \)
61 \( 1 - 3.87T + 61T^{2} \)
67 \( 1 + 0.00393T + 67T^{2} \)
71 \( 1 + 3.59T + 71T^{2} \)
73 \( 1 - 7.82T + 73T^{2} \)
79 \( 1 - 3.03T + 79T^{2} \)
83 \( 1 - 12.5T + 83T^{2} \)
89 \( 1 + 15.2T + 89T^{2} \)
97 \( 1 + 9.71T + 97T^{2} \)
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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

−7.58388799353216636338528020802, −6.81039609155612411905967265899, −6.43118578706430900935259776431, −5.19574598137202812016696077186, −4.46533990411834888934287106678, −3.82859924588550259795994090635, −3.34686455140572478766511708091, −2.23949710425103708705234735818, −1.30161477152882774194915738861, 0, 1.30161477152882774194915738861, 2.23949710425103708705234735818, 3.34686455140572478766511708091, 3.82859924588550259795994090635, 4.46533990411834888934287106678, 5.19574598137202812016696077186, 6.43118578706430900935259776431, 6.81039609155612411905967265899, 7.58388799353216636338528020802

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