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  + 1.05·3-s − 5-s + 3.74·7-s − 1.88·9-s − 4.91·11-s − 1.25·13-s − 1.05·15-s + 3.28·17-s + 2.70·19-s + 3.95·21-s + 0.0904·23-s + 25-s − 5.16·27-s − 3.78·29-s − 2.75·31-s − 5.19·33-s − 3.74·35-s + 6.68·37-s − 1.32·39-s − 8.86·41-s + 9.41·43-s + 1.88·45-s − 4.56·47-s + 7.02·49-s + 3.46·51-s − 12.5·53-s + 4.91·55-s + ⋯
L(s)  = 1  + 0.610·3-s − 0.447·5-s + 1.41·7-s − 0.627·9-s − 1.48·11-s − 0.346·13-s − 0.272·15-s + 0.796·17-s + 0.621·19-s + 0.863·21-s + 0.0188·23-s + 0.200·25-s − 0.993·27-s − 0.703·29-s − 0.494·31-s − 0.904·33-s − 0.632·35-s + 1.09·37-s − 0.211·39-s − 1.38·41-s + 1.43·43-s + 0.280·45-s − 0.665·47-s + 1.00·49-s + 0.485·51-s − 1.71·53-s + 0.663·55-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 \( 1 - 1.05T + 3T^{2} \)
7 \( 1 - 3.74T + 7T^{2} \)
11 \( 1 + 4.91T + 11T^{2} \)
13 \( 1 + 1.25T + 13T^{2} \)
17 \( 1 - 3.28T + 17T^{2} \)
19 \( 1 - 2.70T + 19T^{2} \)
23 \( 1 - 0.0904T + 23T^{2} \)
29 \( 1 + 3.78T + 29T^{2} \)
31 \( 1 + 2.75T + 31T^{2} \)
37 \( 1 - 6.68T + 37T^{2} \)
41 \( 1 + 8.86T + 41T^{2} \)
43 \( 1 - 9.41T + 43T^{2} \)
47 \( 1 + 4.56T + 47T^{2} \)
53 \( 1 + 12.5T + 53T^{2} \)
59 \( 1 - 6.26T + 59T^{2} \)
61 \( 1 - 5.05T + 61T^{2} \)
67 \( 1 - 2.00T + 67T^{2} \)
71 \( 1 + 3.08T + 71T^{2} \)
73 \( 1 - 1.79T + 73T^{2} \)
79 \( 1 + 6.31T + 79T^{2} \)
83 \( 1 + 14.3T + 83T^{2} \)
89 \( 1 + 6.23T + 89T^{2} \)
97 \( 1 + 1.66T + 97T^{2} \)
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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

−7.68373582642794267743427508996, −7.19125540370754860144078884605, −5.85237851972613626031477898221, −5.30737618310498804879540280099, −4.79055326471501531013496851603, −3.83226849723142827006000868211, −2.98603528430651123072535058592, −2.36980631600260063114855047665, −1.38332726423575101759535163903, 0, 1.38332726423575101759535163903, 2.36980631600260063114855047665, 2.98603528430651123072535058592, 3.83226849723142827006000868211, 4.79055326471501531013496851603, 5.30737618310498804879540280099, 5.85237851972613626031477898221, 7.19125540370754860144078884605, 7.68373582642794267743427508996

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