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 + 2.40·3-s + 4-s + 5-s − 2.40·6-s − 1.75·7-s − 8-s + 2.78·9-s − 10-s − 5.60·11-s + 2.40·12-s − 0.293·13-s + 1.75·14-s + 2.40·15-s + 16-s − 4.41·17-s − 2.78·18-s + 5.95·19-s + 20-s − 4.21·21-s + 5.60·22-s + 0.532·23-s − 2.40·24-s + 25-s + 0.293·26-s − 0.510·27-s − 1.75·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.38·3-s + 0.5·4-s + 0.447·5-s − 0.982·6-s − 0.662·7-s − 0.353·8-s + 0.929·9-s − 0.316·10-s − 1.69·11-s + 0.694·12-s − 0.0813·13-s + 0.468·14-s + 0.621·15-s + 0.250·16-s − 1.07·17-s − 0.657·18-s + 1.36·19-s + 0.223·20-s − 0.919·21-s + 1.19·22-s + 0.110·23-s − 0.491·24-s + 0.200·25-s + 0.0575·26-s − 0.0982·27-s − 0.331·28-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 $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + T \)
5 \( 1 - T \)
401 \( 1 - T \)
good3 \( 1 - 2.40T + 3T^{2} \)
7 \( 1 + 1.75T + 7T^{2} \)
11 \( 1 + 5.60T + 11T^{2} \)
13 \( 1 + 0.293T + 13T^{2} \)
17 \( 1 + 4.41T + 17T^{2} \)
19 \( 1 - 5.95T + 19T^{2} \)
23 \( 1 - 0.532T + 23T^{2} \)
29 \( 1 - 3.48T + 29T^{2} \)
31 \( 1 + 8.31T + 31T^{2} \)
37 \( 1 + 6.82T + 37T^{2} \)
41 \( 1 + 3.49T + 41T^{2} \)
43 \( 1 - 9.84T + 43T^{2} \)
47 \( 1 - 6.31T + 47T^{2} \)
53 \( 1 - 11.0T + 53T^{2} \)
59 \( 1 - 7.19T + 59T^{2} \)
61 \( 1 + 13.8T + 61T^{2} \)
67 \( 1 + 13.3T + 67T^{2} \)
71 \( 1 + 1.33T + 71T^{2} \)
73 \( 1 + 10.1T + 73T^{2} \)
79 \( 1 + 2.23T + 79T^{2} \)
83 \( 1 + 4.18T + 83T^{2} \)
89 \( 1 + 13.2T + 89T^{2} \)
97 \( 1 + 10.0T + 97T^{2} \)
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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

−8.211678961337335656409309938478, −7.38070659388281109361867539431, −7.11865914691713341996201392141, −5.87861973409694904825204872388, −5.22057354457780572662650871267, −3.98137484509200225922348131225, −2.87668056435959380789085571934, −2.69341027358928163045947533213, −1.63766370577317880636364761357, 0, 1.63766370577317880636364761357, 2.69341027358928163045947533213, 2.87668056435959380789085571934, 3.98137484509200225922348131225, 5.22057354457780572662650871267, 5.87861973409694904825204872388, 7.11865914691713341996201392141, 7.38070659388281109361867539431, 8.211678961337335656409309938478

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