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  + 1.95·3-s − 5-s + 2.17·7-s + 0.817·9-s + 1.00·11-s − 0.272·13-s − 1.95·15-s − 5.86·17-s − 8.21·19-s + 4.24·21-s + 4.96·23-s + 25-s − 4.26·27-s − 6.90·29-s + 8.98·31-s + 1.95·33-s − 2.17·35-s − 5.23·37-s − 0.532·39-s + 1.37·41-s + 2.99·43-s − 0.817·45-s − 2.43·47-s − 2.28·49-s − 11.4·51-s + 6.76·53-s − 1.00·55-s + ⋯
L(s)  = 1  + 1.12·3-s − 0.447·5-s + 0.820·7-s + 0.272·9-s + 0.302·11-s − 0.0755·13-s − 0.504·15-s − 1.42·17-s − 1.88·19-s + 0.925·21-s + 1.03·23-s + 0.200·25-s − 0.820·27-s − 1.28·29-s + 1.61·31-s + 0.341·33-s − 0.366·35-s − 0.860·37-s − 0.0852·39-s + 0.214·41-s + 0.457·43-s − 0.121·45-s − 0.355·47-s − 0.326·49-s − 1.60·51-s + 0.929·53-s − 0.135·55-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 \( 1 - 1.95T + 3T^{2} \)
7 \( 1 - 2.17T + 7T^{2} \)
11 \( 1 - 1.00T + 11T^{2} \)
13 \( 1 + 0.272T + 13T^{2} \)
17 \( 1 + 5.86T + 17T^{2} \)
19 \( 1 + 8.21T + 19T^{2} \)
23 \( 1 - 4.96T + 23T^{2} \)
29 \( 1 + 6.90T + 29T^{2} \)
31 \( 1 - 8.98T + 31T^{2} \)
37 \( 1 + 5.23T + 37T^{2} \)
41 \( 1 - 1.37T + 41T^{2} \)
43 \( 1 - 2.99T + 43T^{2} \)
47 \( 1 + 2.43T + 47T^{2} \)
53 \( 1 - 6.76T + 53T^{2} \)
59 \( 1 - 2.44T + 59T^{2} \)
61 \( 1 - 4.43T + 61T^{2} \)
67 \( 1 + 12.9T + 67T^{2} \)
71 \( 1 + 6.91T + 71T^{2} \)
73 \( 1 - 3.98T + 73T^{2} \)
79 \( 1 + 9.61T + 79T^{2} \)
83 \( 1 + 11.8T + 83T^{2} \)
89 \( 1 + 11.9T + 89T^{2} \)
97 \( 1 - 10.3T + 97T^{2} \)
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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

−7.60365457103942018617408041193, −6.94877648407208566178305263667, −6.26858977306430974354660518721, −5.25329145770142855853837343675, −4.32014903744413232905711091980, −4.08967175430309092309497283282, −2.96476225037072613703871563215, −2.31883491047130644061990590221, −1.54219253529821082629997779570, 0, 1.54219253529821082629997779570, 2.31883491047130644061990590221, 2.96476225037072613703871563215, 4.08967175430309092309497283282, 4.32014903744413232905711091980, 5.25329145770142855853837343675, 6.26858977306430974354660518721, 6.94877648407208566178305263667, 7.60365457103942018617408041193

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