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  − 1.58·3-s − 5-s + 3.33·7-s − 0.500·9-s − 4.48·11-s + 5.14·13-s + 1.58·15-s − 2.08·17-s − 1.21·19-s − 5.27·21-s − 6.31·23-s + 25-s + 5.53·27-s − 2.70·29-s + 9.30·31-s + 7.08·33-s − 3.33·35-s − 2.68·37-s − 8.13·39-s + 10.6·41-s + 9.65·43-s + 0.500·45-s − 12.9·47-s + 4.11·49-s + 3.29·51-s − 1.27·53-s + 4.48·55-s + ⋯
L(s)  = 1  − 0.912·3-s − 0.447·5-s + 1.25·7-s − 0.166·9-s − 1.35·11-s + 1.42·13-s + 0.408·15-s − 0.505·17-s − 0.279·19-s − 1.15·21-s − 1.31·23-s + 0.200·25-s + 1.06·27-s − 0.501·29-s + 1.67·31-s + 1.23·33-s − 0.563·35-s − 0.442·37-s − 1.30·39-s + 1.66·41-s + 1.47·43-s + 0.0745·45-s − 1.88·47-s + 0.587·49-s + 0.461·51-s − 0.174·53-s + 0.604·55-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 \( 1 + 1.58T + 3T^{2} \)
7 \( 1 - 3.33T + 7T^{2} \)
11 \( 1 + 4.48T + 11T^{2} \)
13 \( 1 - 5.14T + 13T^{2} \)
17 \( 1 + 2.08T + 17T^{2} \)
19 \( 1 + 1.21T + 19T^{2} \)
23 \( 1 + 6.31T + 23T^{2} \)
29 \( 1 + 2.70T + 29T^{2} \)
31 \( 1 - 9.30T + 31T^{2} \)
37 \( 1 + 2.68T + 37T^{2} \)
41 \( 1 - 10.6T + 41T^{2} \)
43 \( 1 - 9.65T + 43T^{2} \)
47 \( 1 + 12.9T + 47T^{2} \)
53 \( 1 + 1.27T + 53T^{2} \)
59 \( 1 - 6.58T + 59T^{2} \)
61 \( 1 + 13.6T + 61T^{2} \)
67 \( 1 + 3.07T + 67T^{2} \)
71 \( 1 - 6.24T + 71T^{2} \)
73 \( 1 + 5.77T + 73T^{2} \)
79 \( 1 + 4.70T + 79T^{2} \)
83 \( 1 - 8.49T + 83T^{2} \)
89 \( 1 - 4.55T + 89T^{2} \)
97 \( 1 + 14.3T + 97T^{2} \)
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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

−7.69556828341889551619682822232, −6.65017248368899116479020364722, −5.94785655288278307774009100743, −5.51060511255758605010054377389, −4.62078592839357454252972473165, −4.24071094923321419481615787640, −3.07452440650373950369574404391, −2.15319869568746851358864166920, −1.09497315953622543086315223578, 0, 1.09497315953622543086315223578, 2.15319869568746851358864166920, 3.07452440650373950369574404391, 4.24071094923321419481615787640, 4.62078592839357454252972473165, 5.51060511255758605010054377389, 5.94785655288278307774009100743, 6.65017248368899116479020364722, 7.69556828341889551619682822232

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