Properties

Degree 2
Conductor $ 2^{2} \cdot 5 \cdot 401 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.10·3-s − 5-s + 1.97·7-s + 6.62·9-s + 0.185·11-s + 5.99·13-s + 3.10·15-s − 1.15·17-s + 2.61·19-s − 6.11·21-s − 0.915·23-s + 25-s − 11.2·27-s − 5.92·29-s + 1.52·31-s − 0.576·33-s − 1.97·35-s − 4.46·37-s − 18.6·39-s − 9.05·41-s + 1.79·43-s − 6.62·45-s − 4.47·47-s − 3.10·49-s + 3.59·51-s − 12.0·53-s − 0.185·55-s + ⋯
L(s)  = 1  − 1.79·3-s − 0.447·5-s + 0.745·7-s + 2.20·9-s + 0.0560·11-s + 1.66·13-s + 0.800·15-s − 0.280·17-s + 0.599·19-s − 1.33·21-s − 0.190·23-s + 0.200·25-s − 2.16·27-s − 1.10·29-s + 0.274·31-s − 0.100·33-s − 0.333·35-s − 0.733·37-s − 2.97·39-s − 1.41·41-s + 0.273·43-s − 0.987·45-s − 0.653·47-s − 0.444·49-s + 0.503·51-s − 1.65·53-s − 0.0250·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8020\)    =    \(2^{2} \cdot 5 \cdot 401\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{8020} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 8020,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5,\;401\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;5,\;401\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
5 \( 1 + T \)
401 \( 1 - T \)
good3 \( 1 + 3.10T + 3T^{2} \)
7 \( 1 - 1.97T + 7T^{2} \)
11 \( 1 - 0.185T + 11T^{2} \)
13 \( 1 - 5.99T + 13T^{2} \)
17 \( 1 + 1.15T + 17T^{2} \)
19 \( 1 - 2.61T + 19T^{2} \)
23 \( 1 + 0.915T + 23T^{2} \)
29 \( 1 + 5.92T + 29T^{2} \)
31 \( 1 - 1.52T + 31T^{2} \)
37 \( 1 + 4.46T + 37T^{2} \)
41 \( 1 + 9.05T + 41T^{2} \)
43 \( 1 - 1.79T + 43T^{2} \)
47 \( 1 + 4.47T + 47T^{2} \)
53 \( 1 + 12.0T + 53T^{2} \)
59 \( 1 + 1.16T + 59T^{2} \)
61 \( 1 - 9.27T + 61T^{2} \)
67 \( 1 - 12.9T + 67T^{2} \)
71 \( 1 + 4.83T + 71T^{2} \)
73 \( 1 - 3.01T + 73T^{2} \)
79 \( 1 - 6.21T + 79T^{2} \)
83 \( 1 + 9.34T + 83T^{2} \)
89 \( 1 + 17.4T + 89T^{2} \)
97 \( 1 - 4.82T + 97T^{2} \)
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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

−7.32282443363823668185372554249, −6.62150477230213218690702723330, −6.11400976082542517224597473710, −5.33589942542712415073373118265, −4.90585819985941117883777172685, −4.04808378741270566556017316862, −3.42710543846794799904925363503, −1.76464726594903658886193192326, −1.13115990512992049218609362628, 0, 1.13115990512992049218609362628, 1.76464726594903658886193192326, 3.42710543846794799904925363503, 4.04808378741270566556017316862, 4.90585819985941117883777172685, 5.33589942542712415073373118265, 6.11400976082542517224597473710, 6.62150477230213218690702723330, 7.32282443363823668185372554249

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