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  + 3.38·3-s − 5-s − 3.56·7-s + 8.48·9-s − 0.809·11-s − 2.35·13-s − 3.38·15-s − 0.960·17-s − 1.53·19-s − 12.0·21-s + 1.98·23-s + 25-s + 18.5·27-s − 7.90·29-s − 1.33·31-s − 2.74·33-s + 3.56·35-s + 4.01·37-s − 7.96·39-s − 1.24·41-s − 8.20·43-s − 8.48·45-s − 4.10·47-s + 5.71·49-s − 3.25·51-s − 3.52·53-s + 0.809·55-s + ⋯
L(s)  = 1  + 1.95·3-s − 0.447·5-s − 1.34·7-s + 2.82·9-s − 0.243·11-s − 0.652·13-s − 0.875·15-s − 0.232·17-s − 0.351·19-s − 2.63·21-s + 0.413·23-s + 0.200·25-s + 3.57·27-s − 1.46·29-s − 0.239·31-s − 0.477·33-s + 0.602·35-s + 0.660·37-s − 1.27·39-s − 0.194·41-s − 1.25·43-s − 1.26·45-s − 0.598·47-s + 0.816·49-s − 0.455·51-s − 0.484·53-s + 0.109·55-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 \( 1 - 3.38T + 3T^{2} \)
7 \( 1 + 3.56T + 7T^{2} \)
11 \( 1 + 0.809T + 11T^{2} \)
13 \( 1 + 2.35T + 13T^{2} \)
17 \( 1 + 0.960T + 17T^{2} \)
19 \( 1 + 1.53T + 19T^{2} \)
23 \( 1 - 1.98T + 23T^{2} \)
29 \( 1 + 7.90T + 29T^{2} \)
31 \( 1 + 1.33T + 31T^{2} \)
37 \( 1 - 4.01T + 37T^{2} \)
41 \( 1 + 1.24T + 41T^{2} \)
43 \( 1 + 8.20T + 43T^{2} \)
47 \( 1 + 4.10T + 47T^{2} \)
53 \( 1 + 3.52T + 53T^{2} \)
59 \( 1 + 0.147T + 59T^{2} \)
61 \( 1 + 6.49T + 61T^{2} \)
67 \( 1 - 3.20T + 67T^{2} \)
71 \( 1 - 2.13T + 71T^{2} \)
73 \( 1 - 1.71T + 73T^{2} \)
79 \( 1 + 9.57T + 79T^{2} \)
83 \( 1 + 5.07T + 83T^{2} \)
89 \( 1 - 10.5T + 89T^{2} \)
97 \( 1 - 4.21T + 97T^{2} \)
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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

−7.67305418885364256863874531854, −6.93581289940237125773004717208, −6.49813276393336604186705814777, −5.23296731904017608392634646899, −4.30528412394355796523547096170, −3.68702710366768693205654728808, −3.07394329390184130711697057940, −2.51946506479633367962711381341, −1.58257707798888445493154716534, 0, 1.58257707798888445493154716534, 2.51946506479633367962711381341, 3.07394329390184130711697057940, 3.68702710366768693205654728808, 4.30528412394355796523547096170, 5.23296731904017608392634646899, 6.49813276393336604186705814777, 6.93581289940237125773004717208, 7.67305418885364256863874531854

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