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  + 0.0547·3-s − 5-s + 3.87·7-s − 2.99·9-s + 3.25·11-s − 6.91·13-s − 0.0547·15-s + 0.379·17-s + 4.37·19-s + 0.212·21-s + 1.01·23-s + 25-s − 0.328·27-s − 3.37·29-s − 6.22·31-s + 0.178·33-s − 3.87·35-s − 0.791·37-s − 0.378·39-s + 2.36·41-s + 0.241·43-s + 2.99·45-s − 6.57·47-s + 7.99·49-s + 0.0208·51-s + 9.52·53-s − 3.25·55-s + ⋯
L(s)  = 1  + 0.0316·3-s − 0.447·5-s + 1.46·7-s − 0.998·9-s + 0.980·11-s − 1.91·13-s − 0.0141·15-s + 0.0921·17-s + 1.00·19-s + 0.0462·21-s + 0.211·23-s + 0.200·25-s − 0.0632·27-s − 0.627·29-s − 1.11·31-s + 0.0310·33-s − 0.654·35-s − 0.130·37-s − 0.0606·39-s + 0.369·41-s + 0.0368·43-s + 0.446·45-s − 0.959·47-s + 1.14·49-s + 0.00291·51-s + 1.30·53-s − 0.438·55-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 \( 1 - 0.0547T + 3T^{2} \)
7 \( 1 - 3.87T + 7T^{2} \)
11 \( 1 - 3.25T + 11T^{2} \)
13 \( 1 + 6.91T + 13T^{2} \)
17 \( 1 - 0.379T + 17T^{2} \)
19 \( 1 - 4.37T + 19T^{2} \)
23 \( 1 - 1.01T + 23T^{2} \)
29 \( 1 + 3.37T + 29T^{2} \)
31 \( 1 + 6.22T + 31T^{2} \)
37 \( 1 + 0.791T + 37T^{2} \)
41 \( 1 - 2.36T + 41T^{2} \)
43 \( 1 - 0.241T + 43T^{2} \)
47 \( 1 + 6.57T + 47T^{2} \)
53 \( 1 - 9.52T + 53T^{2} \)
59 \( 1 - 1.22T + 59T^{2} \)
61 \( 1 + 9.03T + 61T^{2} \)
67 \( 1 + 12.4T + 67T^{2} \)
71 \( 1 + 11.8T + 71T^{2} \)
73 \( 1 + 1.03T + 73T^{2} \)
79 \( 1 - 1.42T + 79T^{2} \)
83 \( 1 - 10.0T + 83T^{2} \)
89 \( 1 + 0.256T + 89T^{2} \)
97 \( 1 - 4.05T + 97T^{2} \)
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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

−7.55063401835874206597012630623, −7.07696309263927805614750531015, −5.97836455741742592793187300003, −5.21166649420376062515359871240, −4.82227355615941657361225794810, −3.96337510379914350410800905605, −3.07214250798099200930426321248, −2.20994468572385158393473208100, −1.31455535381201338664025949617, 0, 1.31455535381201338664025949617, 2.20994468572385158393473208100, 3.07214250798099200930426321248, 3.96337510379914350410800905605, 4.82227355615941657361225794810, 5.21166649420376062515359871240, 5.97836455741742592793187300003, 7.07696309263927805614750531015, 7.55063401835874206597012630623

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