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 + 4.22·5-s + 0.695·7-s + 9-s + 0.705·11-s + 2.93·13-s + 4.22·15-s − 6.80·17-s + 1.49·19-s + 0.695·21-s + 23-s + 12.8·25-s + 27-s − 29-s − 9.11·31-s + 0.705·33-s + 2.93·35-s + 10.9·37-s + 2.93·39-s − 2.16·41-s + 10.9·43-s + 4.22·45-s − 2.16·47-s − 6.51·49-s − 6.80·51-s − 1.12·53-s + 2.98·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.89·5-s + 0.262·7-s + 0.333·9-s + 0.212·11-s + 0.812·13-s + 1.09·15-s − 1.64·17-s + 0.342·19-s + 0.151·21-s + 0.208·23-s + 2.57·25-s + 0.192·27-s − 0.185·29-s − 1.63·31-s + 0.122·33-s + 0.496·35-s + 1.79·37-s + 0.469·39-s − 0.338·41-s + 1.66·43-s + 0.630·45-s − 0.316·47-s − 0.930·49-s − 0.952·51-s − 0.154·53-s + 0.402·55-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  =  $(2,\ 8004,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $4.306841290$
$L(\frac12)$  $\approx$  $4.306841290$
$L(\frac{3}{2})$   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 \( 1 - 4.22T + 5T^{2} \)
7 \( 1 - 0.695T + 7T^{2} \)
11 \( 1 - 0.705T + 11T^{2} \)
13 \( 1 - 2.93T + 13T^{2} \)
17 \( 1 + 6.80T + 17T^{2} \)
19 \( 1 - 1.49T + 19T^{2} \)
31 \( 1 + 9.11T + 31T^{2} \)
37 \( 1 - 10.9T + 37T^{2} \)
41 \( 1 + 2.16T + 41T^{2} \)
43 \( 1 - 10.9T + 43T^{2} \)
47 \( 1 + 2.16T + 47T^{2} \)
53 \( 1 + 1.12T + 53T^{2} \)
59 \( 1 + 5.06T + 59T^{2} \)
61 \( 1 - 13.4T + 61T^{2} \)
67 \( 1 - 2.02T + 67T^{2} \)
71 \( 1 + 11.3T + 71T^{2} \)
73 \( 1 - 9.21T + 73T^{2} \)
79 \( 1 - 2.34T + 79T^{2} \)
83 \( 1 - 12.9T + 83T^{2} \)
89 \( 1 - 6.64T + 89T^{2} \)
97 \( 1 - 9.49T + 97T^{2} \)
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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

−7.897013037358183163482486835639, −7.01697492393482170058513286054, −6.37594148286299348323909549287, −5.87517744660684217852037490400, −5.09130811508137434868690613865, −4.34098789194273704741618899106, −3.39249512408952437386362823836, −2.39449186461298723136613213222, −1.96780024117442564622477502987, −1.06222629920341750577963249055, 1.06222629920341750577963249055, 1.96780024117442564622477502987, 2.39449186461298723136613213222, 3.39249512408952437386362823836, 4.34098789194273704741618899106, 5.09130811508137434868690613865, 5.87517744660684217852037490400, 6.37594148286299348323909549287, 7.01697492393482170058513286054, 7.897013037358183163482486835639

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