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 − 0.154·3-s + 4-s − 5-s + 0.154·6-s − 4.41·7-s − 8-s − 2.97·9-s + 10-s − 3.07·11-s − 0.154·12-s + 6.08·13-s + 4.41·14-s + 0.154·15-s + 16-s + 6.99·17-s + 2.97·18-s − 0.745·19-s − 20-s + 0.683·21-s + 3.07·22-s − 2.22·23-s + 0.154·24-s + 25-s − 6.08·26-s + 0.925·27-s − 4.41·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.0893·3-s + 0.5·4-s − 0.447·5-s + 0.0631·6-s − 1.66·7-s − 0.353·8-s − 0.992·9-s + 0.316·10-s − 0.926·11-s − 0.0446·12-s + 1.68·13-s + 1.18·14-s + 0.0399·15-s + 0.250·16-s + 1.69·17-s + 0.701·18-s − 0.171·19-s − 0.223·20-s + 0.149·21-s + 0.654·22-s − 0.463·23-s + 0.0315·24-s + 0.200·25-s − 1.19·26-s + 0.178·27-s − 0.834·28-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 \( 1 + 0.154T + 3T^{2} \)
7 \( 1 + 4.41T + 7T^{2} \)
11 \( 1 + 3.07T + 11T^{2} \)
13 \( 1 - 6.08T + 13T^{2} \)
17 \( 1 - 6.99T + 17T^{2} \)
19 \( 1 + 0.745T + 19T^{2} \)
23 \( 1 + 2.22T + 23T^{2} \)
29 \( 1 - 2.48T + 29T^{2} \)
31 \( 1 - 2.52T + 31T^{2} \)
37 \( 1 + 3.01T + 37T^{2} \)
41 \( 1 - 6.07T + 41T^{2} \)
43 \( 1 - 1.67T + 43T^{2} \)
47 \( 1 - 2.53T + 47T^{2} \)
53 \( 1 + 0.745T + 53T^{2} \)
59 \( 1 + 3.79T + 59T^{2} \)
61 \( 1 - 4.16T + 61T^{2} \)
67 \( 1 + 13.0T + 67T^{2} \)
71 \( 1 - 8.13T + 71T^{2} \)
73 \( 1 - 1.02T + 73T^{2} \)
79 \( 1 + 7.95T + 79T^{2} \)
83 \( 1 + 6.71T + 83T^{2} \)
89 \( 1 - 8.62T + 89T^{2} \)
97 \( 1 + 16.4T + 97T^{2} \)
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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

−8.166745841704654286958969829535, −7.52726226094679819796355127152, −6.54778891767769567222368734758, −5.98037649564794678744286078589, −5.46417807191464345466154940162, −3.93932781410379138043220494927, −3.21750902013406720900140277475, −2.70165191426004538786269852196, −1.04964725766603752685732044786, 0, 1.04964725766603752685732044786, 2.70165191426004538786269852196, 3.21750902013406720900140277475, 3.93932781410379138043220494927, 5.46417807191464345466154940162, 5.98037649564794678744286078589, 6.54778891767769567222368734758, 7.52726226094679819796355127152, 8.166745841704654286958969829535

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