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 + 2.74·5-s − 3.17·7-s + 9-s − 0.319·11-s − 0.778·13-s + 2.74·15-s − 1.46·17-s − 0.913·19-s − 3.17·21-s + 23-s + 2.51·25-s + 27-s + 29-s − 0.759·31-s − 0.319·33-s − 8.69·35-s − 7.77·37-s − 0.778·39-s − 10.1·41-s − 4.11·43-s + 2.74·45-s − 0.0564·47-s + 3.05·49-s − 1.46·51-s − 11.1·53-s − 0.876·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.22·5-s − 1.19·7-s + 0.333·9-s − 0.0964·11-s − 0.215·13-s + 0.707·15-s − 0.356·17-s − 0.209·19-s − 0.691·21-s + 0.208·23-s + 0.502·25-s + 0.192·27-s + 0.185·29-s − 0.136·31-s − 0.0556·33-s − 1.46·35-s − 1.27·37-s − 0.124·39-s − 1.58·41-s − 0.628·43-s + 0.408·45-s − 0.00822·47-s + 0.436·49-s − 0.205·51-s − 1.53·53-s − 0.118·55-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 \( 1 - 2.74T + 5T^{2} \)
7 \( 1 + 3.17T + 7T^{2} \)
11 \( 1 + 0.319T + 11T^{2} \)
13 \( 1 + 0.778T + 13T^{2} \)
17 \( 1 + 1.46T + 17T^{2} \)
19 \( 1 + 0.913T + 19T^{2} \)
31 \( 1 + 0.759T + 31T^{2} \)
37 \( 1 + 7.77T + 37T^{2} \)
41 \( 1 + 10.1T + 41T^{2} \)
43 \( 1 + 4.11T + 43T^{2} \)
47 \( 1 + 0.0564T + 47T^{2} \)
53 \( 1 + 11.1T + 53T^{2} \)
59 \( 1 - 11.6T + 59T^{2} \)
61 \( 1 - 10.6T + 61T^{2} \)
67 \( 1 - 0.276T + 67T^{2} \)
71 \( 1 + 8.70T + 71T^{2} \)
73 \( 1 + 11.0T + 73T^{2} \)
79 \( 1 + 4.66T + 79T^{2} \)
83 \( 1 - 1.81T + 83T^{2} \)
89 \( 1 + 15.6T + 89T^{2} \)
97 \( 1 - 9.80T + 97T^{2} \)
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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

−7.29334997218066922967253578721, −6.77149717608337747200180513653, −6.21007157514714706555251559338, −5.46495233817822642390302080411, −4.74193935426563392735683496877, −3.67327019004797114166716995331, −3.05436786563321344329255638435, −2.26887277563603058996729836410, −1.51664714391710664031849748978, 0, 1.51664714391710664031849748978, 2.26887277563603058996729836410, 3.05436786563321344329255638435, 3.67327019004797114166716995331, 4.74193935426563392735683496877, 5.46495233817822642390302080411, 6.21007157514714706555251559338, 6.77149717608337747200180513653, 7.29334997218066922967253578721

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