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  + 1.78·3-s − 5-s − 3.35·7-s + 0.193·9-s + 3.67·11-s − 2.18·13-s − 1.78·15-s + 2.43·17-s + 0.922·19-s − 5.98·21-s − 0.388·23-s + 25-s − 5.01·27-s − 3.57·29-s + 4.21·31-s + 6.55·33-s + 3.35·35-s − 7.05·37-s − 3.90·39-s + 7.95·41-s + 8.97·43-s − 0.193·45-s + 0.141·47-s + 4.22·49-s + 4.34·51-s + 13.4·53-s − 3.67·55-s + ⋯
L(s)  = 1  + 1.03·3-s − 0.447·5-s − 1.26·7-s + 0.0644·9-s + 1.10·11-s − 0.605·13-s − 0.461·15-s + 0.590·17-s + 0.211·19-s − 1.30·21-s − 0.0809·23-s + 0.200·25-s − 0.965·27-s − 0.663·29-s + 0.757·31-s + 1.14·33-s + 0.566·35-s − 1.15·37-s − 0.624·39-s + 1.24·41-s + 1.36·43-s − 0.0288·45-s + 0.0205·47-s + 0.603·49-s + 0.608·51-s + 1.84·53-s − 0.494·55-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 \( 1 - 1.78T + 3T^{2} \)
7 \( 1 + 3.35T + 7T^{2} \)
11 \( 1 - 3.67T + 11T^{2} \)
13 \( 1 + 2.18T + 13T^{2} \)
17 \( 1 - 2.43T + 17T^{2} \)
19 \( 1 - 0.922T + 19T^{2} \)
23 \( 1 + 0.388T + 23T^{2} \)
29 \( 1 + 3.57T + 29T^{2} \)
31 \( 1 - 4.21T + 31T^{2} \)
37 \( 1 + 7.05T + 37T^{2} \)
41 \( 1 - 7.95T + 41T^{2} \)
43 \( 1 - 8.97T + 43T^{2} \)
47 \( 1 - 0.141T + 47T^{2} \)
53 \( 1 - 13.4T + 53T^{2} \)
59 \( 1 + 8.54T + 59T^{2} \)
61 \( 1 + 5.52T + 61T^{2} \)
67 \( 1 + 1.45T + 67T^{2} \)
71 \( 1 + 0.463T + 71T^{2} \)
73 \( 1 + 6.01T + 73T^{2} \)
79 \( 1 + 16.8T + 79T^{2} \)
83 \( 1 + 13.6T + 83T^{2} \)
89 \( 1 + 3.56T + 89T^{2} \)
97 \( 1 + 4.44T + 97T^{2} \)
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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

−7.40045320060399566216289690748, −7.05680533811417537708581280789, −6.11819410578749665747933099156, −5.55908418739605281215229621698, −4.29232958078201194481316523881, −3.81470388429781375975068703813, −3.05512279963803379211902316583, −2.55689219698127879998173554844, −1.31341892314447239851968841596, 0, 1.31341892314447239851968841596, 2.55689219698127879998173554844, 3.05512279963803379211902316583, 3.81470388429781375975068703813, 4.29232958078201194481316523881, 5.55908418739605281215229621698, 6.11819410578749665747933099156, 7.05680533811417537708581280789, 7.40045320060399566216289690748

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