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 + 1.72·3-s + 4-s − 5-s − 1.72·6-s − 1.94·7-s − 8-s − 0.0274·9-s + 10-s + 1.83·11-s + 1.72·12-s − 2.05·13-s + 1.94·14-s − 1.72·15-s + 16-s + 1.66·17-s + 0.0274·18-s + 1.54·19-s − 20-s − 3.35·21-s − 1.83·22-s + 0.544·23-s − 1.72·24-s + 25-s + 2.05·26-s − 5.21·27-s − 1.94·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.995·3-s + 0.5·4-s − 0.447·5-s − 0.703·6-s − 0.735·7-s − 0.353·8-s − 0.00913·9-s + 0.316·10-s + 0.554·11-s + 0.497·12-s − 0.568·13-s + 0.519·14-s − 0.445·15-s + 0.250·16-s + 0.402·17-s + 0.00646·18-s + 0.353·19-s − 0.223·20-s − 0.731·21-s − 0.391·22-s + 0.113·23-s − 0.351·24-s + 0.200·25-s + 0.402·26-s − 1.00·27-s − 0.367·28-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 \( 1 - 1.72T + 3T^{2} \)
7 \( 1 + 1.94T + 7T^{2} \)
11 \( 1 - 1.83T + 11T^{2} \)
13 \( 1 + 2.05T + 13T^{2} \)
17 \( 1 - 1.66T + 17T^{2} \)
19 \( 1 - 1.54T + 19T^{2} \)
23 \( 1 - 0.544T + 23T^{2} \)
29 \( 1 - 1.03T + 29T^{2} \)
31 \( 1 - 1.23T + 31T^{2} \)
37 \( 1 - 8.44T + 37T^{2} \)
41 \( 1 - 1.18T + 41T^{2} \)
43 \( 1 + 10.8T + 43T^{2} \)
47 \( 1 + 9.33T + 47T^{2} \)
53 \( 1 - 3.77T + 53T^{2} \)
59 \( 1 + 3.94T + 59T^{2} \)
61 \( 1 - 4.41T + 61T^{2} \)
67 \( 1 + 8.02T + 67T^{2} \)
71 \( 1 + 10.8T + 71T^{2} \)
73 \( 1 + 2.02T + 73T^{2} \)
79 \( 1 - 16.4T + 79T^{2} \)
83 \( 1 + 5.10T + 83T^{2} \)
89 \( 1 - 0.564T + 89T^{2} \)
97 \( 1 + 5.43T + 97T^{2} \)
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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

−8.123404022993115673781333244535, −7.61089342630755579544395547276, −6.79912094810344958750562081255, −6.13978902767488115479052318310, −5.05949688379135110524027083715, −3.95286089836457286264965169109, −3.17920460371468431714458034445, −2.61842058068853548103788715576, −1.41062629219028943355932281330, 0, 1.41062629219028943355932281330, 2.61842058068853548103788715576, 3.17920460371468431714458034445, 3.95286089836457286264965169109, 5.05949688379135110524027083715, 6.13978902767488115479052318310, 6.79912094810344958750562081255, 7.61089342630755579544395547276, 8.123404022993115673781333244535

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