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 − 0.434·5-s − 2.25·7-s + 9-s + 3.49·11-s + 5.02·13-s − 0.434·15-s − 0.922·17-s + 5.87·19-s − 2.25·21-s + 23-s − 4.81·25-s + 27-s − 29-s + 9.57·31-s + 3.49·33-s + 0.980·35-s + 2.89·37-s + 5.02·39-s − 5.70·41-s + 4.57·43-s − 0.434·45-s − 1.28·47-s − 1.89·49-s − 0.922·51-s − 9.88·53-s − 1.51·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.194·5-s − 0.854·7-s + 0.333·9-s + 1.05·11-s + 1.39·13-s − 0.112·15-s − 0.223·17-s + 1.34·19-s − 0.493·21-s + 0.208·23-s − 0.962·25-s + 0.192·27-s − 0.185·29-s + 1.71·31-s + 0.607·33-s + 0.165·35-s + 0.475·37-s + 0.804·39-s − 0.890·41-s + 0.697·43-s − 0.0646·45-s − 0.187·47-s − 0.270·49-s − 0.129·51-s − 1.35·53-s − 0.204·55-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  =  $(2,\ 8004,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.725743429$
$L(\frac12)$  $\approx$  $2.725743429$
$L(\frac{3}{2})$   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 \( 1 + 0.434T + 5T^{2} \)
7 \( 1 + 2.25T + 7T^{2} \)
11 \( 1 - 3.49T + 11T^{2} \)
13 \( 1 - 5.02T + 13T^{2} \)
17 \( 1 + 0.922T + 17T^{2} \)
19 \( 1 - 5.87T + 19T^{2} \)
31 \( 1 - 9.57T + 31T^{2} \)
37 \( 1 - 2.89T + 37T^{2} \)
41 \( 1 + 5.70T + 41T^{2} \)
43 \( 1 - 4.57T + 43T^{2} \)
47 \( 1 + 1.28T + 47T^{2} \)
53 \( 1 + 9.88T + 53T^{2} \)
59 \( 1 + 0.268T + 59T^{2} \)
61 \( 1 + 4.59T + 61T^{2} \)
67 \( 1 - 12.9T + 67T^{2} \)
71 \( 1 + 1.25T + 71T^{2} \)
73 \( 1 + 2.10T + 73T^{2} \)
79 \( 1 + 9.21T + 79T^{2} \)
83 \( 1 - 3.24T + 83T^{2} \)
89 \( 1 - 8.36T + 89T^{2} \)
97 \( 1 - 0.502T + 97T^{2} \)
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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

−7.899852322923072769387802964487, −7.15513933908983826512302344815, −6.33110379637867745402535283108, −6.09243850541430237025173251746, −4.92871841899464999249192893905, −4.04790523188962740822310065469, −3.47921519907534649883109069844, −2.92213407385739563599553695712, −1.69446228610328820895965450073, −0.840156139820717569782745286461, 0.840156139820717569782745286461, 1.69446228610328820895965450073, 2.92213407385739563599553695712, 3.47921519907534649883109069844, 4.04790523188962740822310065469, 4.92871841899464999249192893905, 6.09243850541430237025173251746, 6.33110379637867745402535283108, 7.15513933908983826512302344815, 7.899852322923072769387802964487

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