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 − 3.12·5-s − 0.760·7-s + 9-s + 3.50·11-s − 3.64·13-s − 3.12·15-s + 6.89·17-s − 2.74·19-s − 0.760·21-s + 23-s + 4.73·25-s + 27-s − 29-s − 5.32·31-s + 3.50·33-s + 2.37·35-s + 5.93·37-s − 3.64·39-s + 6.52·41-s − 1.04·43-s − 3.12·45-s − 8.77·47-s − 6.42·49-s + 6.89·51-s + 6.18·53-s − 10.9·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.39·5-s − 0.287·7-s + 0.333·9-s + 1.05·11-s − 1.01·13-s − 0.805·15-s + 1.67·17-s − 0.629·19-s − 0.166·21-s + 0.208·23-s + 0.947·25-s + 0.192·27-s − 0.185·29-s − 0.955·31-s + 0.610·33-s + 0.401·35-s + 0.975·37-s − 0.584·39-s + 1.01·41-s − 0.158·43-s − 0.465·45-s − 1.28·47-s − 0.917·49-s + 0.965·51-s + 0.849·53-s − 1.47·55-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  =  $(2,\ 8004,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.668015349$
$L(\frac12)$  $\approx$  $1.668015349$
$L(\frac{3}{2})$   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 \( 1 + 3.12T + 5T^{2} \)
7 \( 1 + 0.760T + 7T^{2} \)
11 \( 1 - 3.50T + 11T^{2} \)
13 \( 1 + 3.64T + 13T^{2} \)
17 \( 1 - 6.89T + 17T^{2} \)
19 \( 1 + 2.74T + 19T^{2} \)
31 \( 1 + 5.32T + 31T^{2} \)
37 \( 1 - 5.93T + 37T^{2} \)
41 \( 1 - 6.52T + 41T^{2} \)
43 \( 1 + 1.04T + 43T^{2} \)
47 \( 1 + 8.77T + 47T^{2} \)
53 \( 1 - 6.18T + 53T^{2} \)
59 \( 1 + 11.2T + 59T^{2} \)
61 \( 1 - 0.576T + 61T^{2} \)
67 \( 1 - 3.98T + 67T^{2} \)
71 \( 1 + 6.42T + 71T^{2} \)
73 \( 1 - 7.28T + 73T^{2} \)
79 \( 1 - 6.45T + 79T^{2} \)
83 \( 1 - 2.58T + 83T^{2} \)
89 \( 1 + 10.7T + 89T^{2} \)
97 \( 1 - 17.1T + 97T^{2} \)
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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

−7.84664518345661065274140297973, −7.31793981635778601060951185212, −6.67243015135006518919027970485, −5.78453816490652889158964905129, −4.80258873514665009454739882651, −4.14226166182685167261323378297, −3.50943630416459516371677093386, −2.94383304722950697747091844021, −1.76372631882040785730639114767, −0.62398524502311777304020449595, 0.62398524502311777304020449595, 1.76372631882040785730639114767, 2.94383304722950697747091844021, 3.50943630416459516371677093386, 4.14226166182685167261323378297, 4.80258873514665009454739882651, 5.78453816490652889158964905129, 6.67243015135006518919027970485, 7.31793981635778601060951185212, 7.84664518345661065274140297973

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