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  + 0.356·3-s − 5-s − 0.400·7-s − 2.87·9-s − 0.436·11-s − 0.615·13-s − 0.356·15-s − 0.540·17-s + 1.98·19-s − 0.142·21-s + 0.0321·23-s + 25-s − 2.09·27-s + 7.32·29-s − 1.85·31-s − 0.155·33-s + 0.400·35-s + 1.34·37-s − 0.219·39-s + 7.68·41-s + 4.12·43-s + 2.87·45-s + 11.3·47-s − 6.83·49-s − 0.192·51-s − 3.77·53-s + 0.436·55-s + ⋯
L(s)  = 1  + 0.205·3-s − 0.447·5-s − 0.151·7-s − 0.957·9-s − 0.131·11-s − 0.170·13-s − 0.0919·15-s − 0.131·17-s + 0.454·19-s − 0.0311·21-s + 0.00669·23-s + 0.200·25-s − 0.402·27-s + 1.36·29-s − 0.332·31-s − 0.0270·33-s + 0.0676·35-s + 0.220·37-s − 0.0350·39-s + 1.20·41-s + 0.629·43-s + 0.428·45-s + 1.64·47-s − 0.977·49-s − 0.0269·51-s − 0.518·53-s + 0.0588·55-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 \( 1 - 0.356T + 3T^{2} \)
7 \( 1 + 0.400T + 7T^{2} \)
11 \( 1 + 0.436T + 11T^{2} \)
13 \( 1 + 0.615T + 13T^{2} \)
17 \( 1 + 0.540T + 17T^{2} \)
19 \( 1 - 1.98T + 19T^{2} \)
23 \( 1 - 0.0321T + 23T^{2} \)
29 \( 1 - 7.32T + 29T^{2} \)
31 \( 1 + 1.85T + 31T^{2} \)
37 \( 1 - 1.34T + 37T^{2} \)
41 \( 1 - 7.68T + 41T^{2} \)
43 \( 1 - 4.12T + 43T^{2} \)
47 \( 1 - 11.3T + 47T^{2} \)
53 \( 1 + 3.77T + 53T^{2} \)
59 \( 1 - 3.82T + 59T^{2} \)
61 \( 1 + 1.59T + 61T^{2} \)
67 \( 1 + 10.1T + 67T^{2} \)
71 \( 1 + 8.53T + 71T^{2} \)
73 \( 1 - 1.94T + 73T^{2} \)
79 \( 1 + 2.05T + 79T^{2} \)
83 \( 1 - 3.42T + 83T^{2} \)
89 \( 1 + 9.88T + 89T^{2} \)
97 \( 1 + 4.60T + 97T^{2} \)
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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

−7.60173782168034794738494038903, −6.85703292406759448840534925124, −6.05211650897948098090098567800, −5.46512831414042718680821100816, −4.60129753134544161179429158283, −3.89957941152044495722504266075, −2.95026025589941383464343444846, −2.52186563392001500761053665671, −1.16051571896793194894995080154, 0, 1.16051571896793194894995080154, 2.52186563392001500761053665671, 2.95026025589941383464343444846, 3.89957941152044495722504266075, 4.60129753134544161179429158283, 5.46512831414042718680821100816, 6.05211650897948098090098567800, 6.85703292406759448840534925124, 7.60173782168034794738494038903

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