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 − 3.41·5-s + 2.05·7-s + 9-s + 1.14·11-s − 4.51·13-s − 3.41·15-s + 0.446·17-s − 2.11·19-s + 2.05·21-s + 23-s + 6.67·25-s + 27-s + 29-s + 6.57·31-s + 1.14·33-s − 7.00·35-s − 4.89·37-s − 4.51·39-s + 3.92·41-s + 3.16·43-s − 3.41·45-s − 0.451·47-s − 2.79·49-s + 0.446·51-s + 4.11·53-s − 3.91·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.52·5-s + 0.775·7-s + 0.333·9-s + 0.345·11-s − 1.25·13-s − 0.882·15-s + 0.108·17-s − 0.484·19-s + 0.447·21-s + 0.208·23-s + 1.33·25-s + 0.192·27-s + 0.185·29-s + 1.18·31-s + 0.199·33-s − 1.18·35-s − 0.804·37-s − 0.722·39-s + 0.613·41-s + 0.482·43-s − 0.509·45-s − 0.0659·47-s − 0.399·49-s + 0.0625·51-s + 0.565·53-s − 0.528·55-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 \( 1 + 3.41T + 5T^{2} \)
7 \( 1 - 2.05T + 7T^{2} \)
11 \( 1 - 1.14T + 11T^{2} \)
13 \( 1 + 4.51T + 13T^{2} \)
17 \( 1 - 0.446T + 17T^{2} \)
19 \( 1 + 2.11T + 19T^{2} \)
31 \( 1 - 6.57T + 31T^{2} \)
37 \( 1 + 4.89T + 37T^{2} \)
41 \( 1 - 3.92T + 41T^{2} \)
43 \( 1 - 3.16T + 43T^{2} \)
47 \( 1 + 0.451T + 47T^{2} \)
53 \( 1 - 4.11T + 53T^{2} \)
59 \( 1 + 6.31T + 59T^{2} \)
61 \( 1 + 8.75T + 61T^{2} \)
67 \( 1 + 10.6T + 67T^{2} \)
71 \( 1 - 13.3T + 71T^{2} \)
73 \( 1 + 12.6T + 73T^{2} \)
79 \( 1 - 2.30T + 79T^{2} \)
83 \( 1 + 4.59T + 83T^{2} \)
89 \( 1 - 14.8T + 89T^{2} \)
97 \( 1 - 8.14T + 97T^{2} \)
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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

−7.70073081501575261274092537474, −7.07063299438342698538288053514, −6.27624597197107371817504220765, −5.05402665494948295273218625737, −4.56328685496282345329244364101, −3.98571072462752033294132964181, −3.13224947603627666351764390864, −2.36899913905120411432138558132, −1.23806246603826155256980515818, 0, 1.23806246603826155256980515818, 2.36899913905120411432138558132, 3.13224947603627666351764390864, 3.98571072462752033294132964181, 4.56328685496282345329244364101, 5.05402665494948295273218625737, 6.27624597197107371817504220765, 7.07063299438342698538288053514, 7.70073081501575261274092537474

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