Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 23 \cdot 29 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 2.96·5-s − 3.61·7-s + 9-s − 1.51·11-s + 0.595·13-s + 2.96·15-s + 0.748·17-s − 3.13·19-s − 3.61·21-s + 23-s + 3.81·25-s + 27-s − 29-s + 1.50·31-s − 1.51·33-s − 10.7·35-s + 3.65·37-s + 0.595·39-s − 0.901·41-s − 7.23·43-s + 2.96·45-s + 6.53·47-s + 6.07·49-s + 0.748·51-s + 11.1·53-s − 4.50·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.32·5-s − 1.36·7-s + 0.333·9-s − 0.457·11-s + 0.165·13-s + 0.766·15-s + 0.181·17-s − 0.719·19-s − 0.788·21-s + 0.208·23-s + 0.763·25-s + 0.192·27-s − 0.185·29-s + 0.270·31-s − 0.264·33-s − 1.81·35-s + 0.601·37-s + 0.0954·39-s − 0.140·41-s − 1.10·43-s + 0.442·45-s + 0.953·47-s + 0.867·49-s + 0.104·51-s + 1.52·53-s − 0.607·55-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  =  0
Selberg data  =  $(2,\ 8004,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.699369504$
$L(\frac12)$  $\approx$  $2.699369504$
$L(\frac{3}{2})$   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 \( 1 - 2.96T + 5T^{2} \)
7 \( 1 + 3.61T + 7T^{2} \)
11 \( 1 + 1.51T + 11T^{2} \)
13 \( 1 - 0.595T + 13T^{2} \)
17 \( 1 - 0.748T + 17T^{2} \)
19 \( 1 + 3.13T + 19T^{2} \)
31 \( 1 - 1.50T + 31T^{2} \)
37 \( 1 - 3.65T + 37T^{2} \)
41 \( 1 + 0.901T + 41T^{2} \)
43 \( 1 + 7.23T + 43T^{2} \)
47 \( 1 - 6.53T + 47T^{2} \)
53 \( 1 - 11.1T + 53T^{2} \)
59 \( 1 - 14.9T + 59T^{2} \)
61 \( 1 - 4.00T + 61T^{2} \)
67 \( 1 - 7.19T + 67T^{2} \)
71 \( 1 - 7.23T + 71T^{2} \)
73 \( 1 - 8.91T + 73T^{2} \)
79 \( 1 - 8.63T + 79T^{2} \)
83 \( 1 + 7.09T + 83T^{2} \)
89 \( 1 - 14.5T + 89T^{2} \)
97 \( 1 + 7.05T + 97T^{2} \)
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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

−7.940729565740858975984604709144, −6.80647526351939962056211456274, −6.65656729723729512419737906733, −5.73693678128754861870327811149, −5.27102979838816971135550929935, −4.11847598261819518822248610575, −3.39824165135558593801879354240, −2.54609175064899504735305159912, −2.07149862211835433337128095158, −0.77111572114126963912441107576, 0.77111572114126963912441107576, 2.07149862211835433337128095158, 2.54609175064899504735305159912, 3.39824165135558593801879354240, 4.11847598261819518822248610575, 5.27102979838816971135550929935, 5.73693678128754861870327811149, 6.65656729723729512419737906733, 6.80647526351939962056211456274, 7.940729565740858975984604709144

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