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  + 0.910·3-s − 5-s + 1.40·7-s − 2.17·9-s − 2.72·11-s + 4.96·13-s − 0.910·15-s + 1.31·17-s − 1.68·19-s + 1.28·21-s + 4.30·23-s + 25-s − 4.70·27-s − 5.98·29-s + 0.173·31-s − 2.48·33-s − 1.40·35-s − 2.08·37-s + 4.52·39-s − 1.57·41-s − 10.3·43-s + 2.17·45-s − 2.61·47-s − 5.01·49-s + 1.20·51-s + 2.01·53-s + 2.72·55-s + ⋯
L(s)  = 1  + 0.525·3-s − 0.447·5-s + 0.532·7-s − 0.723·9-s − 0.821·11-s + 1.37·13-s − 0.234·15-s + 0.319·17-s − 0.385·19-s + 0.279·21-s + 0.897·23-s + 0.200·25-s − 0.905·27-s − 1.11·29-s + 0.0310·31-s − 0.431·33-s − 0.238·35-s − 0.343·37-s + 0.723·39-s − 0.246·41-s − 1.58·43-s + 0.323·45-s − 0.381·47-s − 0.716·49-s + 0.168·51-s + 0.276·53-s + 0.367·55-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 \( 1 - 0.910T + 3T^{2} \)
7 \( 1 - 1.40T + 7T^{2} \)
11 \( 1 + 2.72T + 11T^{2} \)
13 \( 1 - 4.96T + 13T^{2} \)
17 \( 1 - 1.31T + 17T^{2} \)
19 \( 1 + 1.68T + 19T^{2} \)
23 \( 1 - 4.30T + 23T^{2} \)
29 \( 1 + 5.98T + 29T^{2} \)
31 \( 1 - 0.173T + 31T^{2} \)
37 \( 1 + 2.08T + 37T^{2} \)
41 \( 1 + 1.57T + 41T^{2} \)
43 \( 1 + 10.3T + 43T^{2} \)
47 \( 1 + 2.61T + 47T^{2} \)
53 \( 1 - 2.01T + 53T^{2} \)
59 \( 1 - 3.37T + 59T^{2} \)
61 \( 1 - 3.01T + 61T^{2} \)
67 \( 1 + 1.07T + 67T^{2} \)
71 \( 1 - 0.238T + 71T^{2} \)
73 \( 1 + 11.9T + 73T^{2} \)
79 \( 1 - 4.59T + 79T^{2} \)
83 \( 1 - 1.18T + 83T^{2} \)
89 \( 1 - 1.48T + 89T^{2} \)
97 \( 1 + 14.7T + 97T^{2} \)
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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

−7.68785463208037191885336549032, −6.90331717735686120573095189594, −6.04368036606594990428387995898, −5.36749281754589194301201658918, −4.70812687769181539204913726273, −3.63831102074964740974425211632, −3.26727922776621190125701686618, −2.28233717631987251710819881695, −1.34547215080914100432521626491, 0, 1.34547215080914100432521626491, 2.28233717631987251710819881695, 3.26727922776621190125701686618, 3.63831102074964740974425211632, 4.70812687769181539204913726273, 5.36749281754589194301201658918, 6.04368036606594990428387995898, 6.90331717735686120573095189594, 7.68785463208037191885336549032

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