Properties

Degree 2
Conductor $ 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  − 2-s − 3.00·3-s + 4-s − 5-s + 3.00·6-s − 4.38·7-s − 8-s + 6.02·9-s + 10-s − 3.27·11-s − 3.00·12-s − 2.26·13-s + 4.38·14-s + 3.00·15-s + 16-s + 0.368·17-s − 6.02·18-s − 1.86·19-s − 20-s + 13.1·21-s + 3.27·22-s − 0.625·23-s + 3.00·24-s + 25-s + 2.26·26-s − 9.08·27-s − 4.38·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.73·3-s + 0.5·4-s − 0.447·5-s + 1.22·6-s − 1.65·7-s − 0.353·8-s + 2.00·9-s + 0.316·10-s − 0.986·11-s − 0.867·12-s − 0.627·13-s + 1.17·14-s + 0.775·15-s + 0.250·16-s + 0.0893·17-s − 1.42·18-s − 0.427·19-s − 0.223·20-s + 2.87·21-s + 0.697·22-s − 0.130·23-s + 0.613·24-s + 0.200·25-s + 0.443·26-s − 1.74·27-s − 0.827·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4010\)    =    \(2 \cdot 5 \cdot 401\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{4010} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 4010,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 \( 1 + T \)
5 \( 1 + T \)
401 \( 1 + T \)
good3 \( 1 + 3.00T + 3T^{2} \)
7 \( 1 + 4.38T + 7T^{2} \)
11 \( 1 + 3.27T + 11T^{2} \)
13 \( 1 + 2.26T + 13T^{2} \)
17 \( 1 - 0.368T + 17T^{2} \)
19 \( 1 + 1.86T + 19T^{2} \)
23 \( 1 + 0.625T + 23T^{2} \)
29 \( 1 - 0.824T + 29T^{2} \)
31 \( 1 - 2.48T + 31T^{2} \)
37 \( 1 + 3.19T + 37T^{2} \)
41 \( 1 - 4.93T + 41T^{2} \)
43 \( 1 - 5.23T + 43T^{2} \)
47 \( 1 + 1.07T + 47T^{2} \)
53 \( 1 - 0.543T + 53T^{2} \)
59 \( 1 - 1.14T + 59T^{2} \)
61 \( 1 + 6.83T + 61T^{2} \)
67 \( 1 + 4.96T + 67T^{2} \)
71 \( 1 - 5.72T + 71T^{2} \)
73 \( 1 - 16.8T + 73T^{2} \)
79 \( 1 - 8.03T + 79T^{2} \)
83 \( 1 + 14.4T + 83T^{2} \)
89 \( 1 + 0.445T + 89T^{2} \)
97 \( 1 - 13.9T + 97T^{2} \)
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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

−7.85474262355345565903345784939, −7.23956533701473978530388888071, −6.55259753717983899131750908599, −6.05487010139370925468026788957, −5.29386151185845657244935888650, −4.44155730077405645906759724933, −3.37607803324597614670064524724, −2.35887889159300104334416090834, −0.73174624755586267073049002520, 0, 0.73174624755586267073049002520, 2.35887889159300104334416090834, 3.37607803324597614670064524724, 4.44155730077405645906759724933, 5.29386151185845657244935888650, 6.05487010139370925468026788957, 6.55259753717983899131750908599, 7.23956533701473978530388888071, 7.85474262355345565903345784939

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