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 + 0.187·5-s + 3.11·7-s + 9-s + 0.0152·11-s + 5.09·13-s + 0.187·15-s + 2.80·17-s + 4.44·19-s + 3.11·21-s + 23-s − 4.96·25-s + 27-s − 29-s − 3.85·31-s + 0.0152·33-s + 0.584·35-s − 2.44·37-s + 5.09·39-s + 4.07·41-s − 4.10·43-s + 0.187·45-s + 10.7·47-s + 2.67·49-s + 2.80·51-s + 8.69·53-s + 0.00285·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.0839·5-s + 1.17·7-s + 0.333·9-s + 0.00458·11-s + 1.41·13-s + 0.0484·15-s + 0.680·17-s + 1.01·19-s + 0.678·21-s + 0.208·23-s − 0.992·25-s + 0.192·27-s − 0.185·29-s − 0.691·31-s + 0.00264·33-s + 0.0987·35-s − 0.402·37-s + 0.815·39-s + 0.636·41-s − 0.626·43-s + 0.0279·45-s + 1.57·47-s + 0.382·49-s + 0.392·51-s + 1.19·53-s + 0.000385·55-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  =  $(2,\ 8004,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.707201109$
$L(\frac12)$  $\approx$  $3.707201109$
$L(\frac{3}{2})$   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 \( 1 - 0.187T + 5T^{2} \)
7 \( 1 - 3.11T + 7T^{2} \)
11 \( 1 - 0.0152T + 11T^{2} \)
13 \( 1 - 5.09T + 13T^{2} \)
17 \( 1 - 2.80T + 17T^{2} \)
19 \( 1 - 4.44T + 19T^{2} \)
31 \( 1 + 3.85T + 31T^{2} \)
37 \( 1 + 2.44T + 37T^{2} \)
41 \( 1 - 4.07T + 41T^{2} \)
43 \( 1 + 4.10T + 43T^{2} \)
47 \( 1 - 10.7T + 47T^{2} \)
53 \( 1 - 8.69T + 53T^{2} \)
59 \( 1 + 8.68T + 59T^{2} \)
61 \( 1 - 2.99T + 61T^{2} \)
67 \( 1 - 11.2T + 67T^{2} \)
71 \( 1 - 4.10T + 71T^{2} \)
73 \( 1 - 12.5T + 73T^{2} \)
79 \( 1 - 6.51T + 79T^{2} \)
83 \( 1 + 4.47T + 83T^{2} \)
89 \( 1 + 9.25T + 89T^{2} \)
97 \( 1 - 7.78T + 97T^{2} \)
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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

−7.950751711441871236840343301760, −7.35026262660624117654858014416, −6.50629375023270414346000518917, −5.53160723414690057724507891770, −5.22815901785833607640177223158, −4.01991533299310400220237226339, −3.68854011926725265761331073881, −2.63548236034821687033327197991, −1.68835552744437842029053233958, −1.03110399521854255839071144618, 1.03110399521854255839071144618, 1.68835552744437842029053233958, 2.63548236034821687033327197991, 3.68854011926725265761331073881, 4.01991533299310400220237226339, 5.22815901785833607640177223158, 5.53160723414690057724507891770, 6.50629375023270414346000518917, 7.35026262660624117654858014416, 7.950751711441871236840343301760

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