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 + 3.15·3-s + 4-s − 5-s − 3.15·6-s − 0.474·7-s − 8-s + 6.94·9-s + 10-s − 1.05·11-s + 3.15·12-s − 3.43·13-s + 0.474·14-s − 3.15·15-s + 16-s − 4.69·17-s − 6.94·18-s − 6.53·19-s − 20-s − 1.49·21-s + 1.05·22-s − 6.00·23-s − 3.15·24-s + 25-s + 3.43·26-s + 12.4·27-s − 0.474·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.82·3-s + 0.5·4-s − 0.447·5-s − 1.28·6-s − 0.179·7-s − 0.353·8-s + 2.31·9-s + 0.316·10-s − 0.317·11-s + 0.910·12-s − 0.953·13-s + 0.126·14-s − 0.814·15-s + 0.250·16-s − 1.13·17-s − 1.63·18-s − 1.49·19-s − 0.223·20-s − 0.326·21-s + 0.224·22-s − 1.25·23-s − 0.643·24-s + 0.200·25-s + 0.674·26-s + 2.39·27-s − 0.0896·28-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 \( 1 - 3.15T + 3T^{2} \)
7 \( 1 + 0.474T + 7T^{2} \)
11 \( 1 + 1.05T + 11T^{2} \)
13 \( 1 + 3.43T + 13T^{2} \)
17 \( 1 + 4.69T + 17T^{2} \)
19 \( 1 + 6.53T + 19T^{2} \)
23 \( 1 + 6.00T + 23T^{2} \)
29 \( 1 - 8.30T + 29T^{2} \)
31 \( 1 - 0.592T + 31T^{2} \)
37 \( 1 - 0.575T + 37T^{2} \)
41 \( 1 - 2.13T + 41T^{2} \)
43 \( 1 + 3.03T + 43T^{2} \)
47 \( 1 + 1.96T + 47T^{2} \)
53 \( 1 + 6.30T + 53T^{2} \)
59 \( 1 + 4.62T + 59T^{2} \)
61 \( 1 - 13.0T + 61T^{2} \)
67 \( 1 + 14.9T + 67T^{2} \)
71 \( 1 + 10.7T + 71T^{2} \)
73 \( 1 + 6.62T + 73T^{2} \)
79 \( 1 + 10.7T + 79T^{2} \)
83 \( 1 - 12.6T + 83T^{2} \)
89 \( 1 - 2.47T + 89T^{2} \)
97 \( 1 + 7.44T + 97T^{2} \)
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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

−8.205176098652513788995128843087, −7.68537943133778383248273803915, −6.90147921598437379445543909267, −6.28435891466954120568655332072, −4.65036546030646392683856974540, −4.18106735630310969842956073417, −3.08272135971167313871694288390, −2.46401785391805847781229686618, −1.75024833826189812282127020893, 0, 1.75024833826189812282127020893, 2.46401785391805847781229686618, 3.08272135971167313871694288390, 4.18106735630310969842956073417, 4.65036546030646392683856974540, 6.28435891466954120568655332072, 6.90147921598437379445543909267, 7.68537943133778383248273803915, 8.205176098652513788995128843087

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