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  − 1.42·3-s − 5-s − 3.82·7-s − 0.967·9-s + 2.52·11-s + 7.13·13-s + 1.42·15-s − 6.37·17-s − 4.40·19-s + 5.45·21-s + 1.00·23-s + 25-s + 5.65·27-s + 2.93·29-s − 4.16·31-s − 3.59·33-s + 3.82·35-s − 11.7·37-s − 10.1·39-s + 1.67·41-s + 9.49·43-s + 0.967·45-s + 9.50·47-s + 7.64·49-s + 9.09·51-s + 5.55·53-s − 2.52·55-s + ⋯
L(s)  = 1  − 0.823·3-s − 0.447·5-s − 1.44·7-s − 0.322·9-s + 0.760·11-s + 1.98·13-s + 0.368·15-s − 1.54·17-s − 1.01·19-s + 1.19·21-s + 0.210·23-s + 0.200·25-s + 1.08·27-s + 0.545·29-s − 0.747·31-s − 0.625·33-s + 0.646·35-s − 1.93·37-s − 1.62·39-s + 0.261·41-s + 1.44·43-s + 0.144·45-s + 1.38·47-s + 1.09·49-s + 1.27·51-s + 0.763·53-s − 0.340·55-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 \( 1 + 1.42T + 3T^{2} \)
7 \( 1 + 3.82T + 7T^{2} \)
11 \( 1 - 2.52T + 11T^{2} \)
13 \( 1 - 7.13T + 13T^{2} \)
17 \( 1 + 6.37T + 17T^{2} \)
19 \( 1 + 4.40T + 19T^{2} \)
23 \( 1 - 1.00T + 23T^{2} \)
29 \( 1 - 2.93T + 29T^{2} \)
31 \( 1 + 4.16T + 31T^{2} \)
37 \( 1 + 11.7T + 37T^{2} \)
41 \( 1 - 1.67T + 41T^{2} \)
43 \( 1 - 9.49T + 43T^{2} \)
47 \( 1 - 9.50T + 47T^{2} \)
53 \( 1 - 5.55T + 53T^{2} \)
59 \( 1 - 5.03T + 59T^{2} \)
61 \( 1 + 6.50T + 61T^{2} \)
67 \( 1 + 10.7T + 67T^{2} \)
71 \( 1 - 16.2T + 71T^{2} \)
73 \( 1 - 4.97T + 73T^{2} \)
79 \( 1 + 5.11T + 79T^{2} \)
83 \( 1 + 1.92T + 83T^{2} \)
89 \( 1 + 16.4T + 89T^{2} \)
97 \( 1 - 13.7T + 97T^{2} \)
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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

−7.10137916865734974044848963631, −6.70139782990497712926882896955, −6.10040541512899372208114795574, −5.74599585331704623470448774543, −4.53870635383769136518287107121, −3.87295936297229250312135033151, −3.33010614764775513617486291991, −2.25036510544533604110382119752, −0.932539278313875518478159747716, 0, 0.932539278313875518478159747716, 2.25036510544533604110382119752, 3.33010614764775513617486291991, 3.87295936297229250312135033151, 4.53870635383769136518287107121, 5.74599585331704623470448774543, 6.10040541512899372208114795574, 6.70139782990497712926882896955, 7.10137916865734974044848963631

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