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  + 1.28·3-s − 5-s − 0.791·7-s − 1.35·9-s + 5.58·11-s + 0.759·13-s − 1.28·15-s − 7.02·17-s + 4.40·19-s − 1.01·21-s + 3.48·23-s + 25-s − 5.58·27-s − 4.93·29-s − 5.65·31-s + 7.16·33-s + 0.791·35-s + 2.56·37-s + 0.973·39-s − 1.22·41-s − 6.23·43-s + 1.35·45-s − 6.02·47-s − 6.37·49-s − 9.00·51-s − 8.35·53-s − 5.58·55-s + ⋯
L(s)  = 1  + 0.740·3-s − 0.447·5-s − 0.299·7-s − 0.452·9-s + 1.68·11-s + 0.210·13-s − 0.330·15-s − 1.70·17-s + 1.01·19-s − 0.221·21-s + 0.727·23-s + 0.200·25-s − 1.07·27-s − 0.915·29-s − 1.01·31-s + 1.24·33-s + 0.133·35-s + 0.421·37-s + 0.155·39-s − 0.191·41-s − 0.951·43-s + 0.202·45-s − 0.878·47-s − 0.910·49-s − 1.26·51-s − 1.14·53-s − 0.753·55-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 \( 1 - 1.28T + 3T^{2} \)
7 \( 1 + 0.791T + 7T^{2} \)
11 \( 1 - 5.58T + 11T^{2} \)
13 \( 1 - 0.759T + 13T^{2} \)
17 \( 1 + 7.02T + 17T^{2} \)
19 \( 1 - 4.40T + 19T^{2} \)
23 \( 1 - 3.48T + 23T^{2} \)
29 \( 1 + 4.93T + 29T^{2} \)
31 \( 1 + 5.65T + 31T^{2} \)
37 \( 1 - 2.56T + 37T^{2} \)
41 \( 1 + 1.22T + 41T^{2} \)
43 \( 1 + 6.23T + 43T^{2} \)
47 \( 1 + 6.02T + 47T^{2} \)
53 \( 1 + 8.35T + 53T^{2} \)
59 \( 1 + 9.38T + 59T^{2} \)
61 \( 1 - 12.5T + 61T^{2} \)
67 \( 1 - 6.07T + 67T^{2} \)
71 \( 1 - 3.00T + 71T^{2} \)
73 \( 1 - 13.9T + 73T^{2} \)
79 \( 1 + 8.59T + 79T^{2} \)
83 \( 1 - 6.43T + 83T^{2} \)
89 \( 1 + 18.5T + 89T^{2} \)
97 \( 1 - 2.63T + 97T^{2} \)
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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

−7.51045131793989146965755072294, −6.75665200941712133989846717167, −6.36464810650629533202215978785, −5.34596456210836385177626331301, −4.52139320450597536553218189513, −3.61783090159664816789394259696, −3.36540791097433503519797949028, −2.26441017741009549029996490877, −1.38444681704797425284282462107, 0, 1.38444681704797425284282462107, 2.26441017741009549029996490877, 3.36540791097433503519797949028, 3.61783090159664816789394259696, 4.52139320450597536553218189513, 5.34596456210836385177626331301, 6.36464810650629533202215978785, 6.75665200941712133989846717167, 7.51045131793989146965755072294

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