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 + 2.71·3-s + 4-s − 5-s − 2.71·6-s − 0.594·7-s − 8-s + 4.39·9-s + 10-s − 5.15·11-s + 2.71·12-s + 0.430·13-s + 0.594·14-s − 2.71·15-s + 16-s + 3.55·17-s − 4.39·18-s − 5.11·19-s − 20-s − 1.61·21-s + 5.15·22-s + 6.74·23-s − 2.71·24-s + 25-s − 0.430·26-s + 3.79·27-s − 0.594·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.57·3-s + 0.5·4-s − 0.447·5-s − 1.11·6-s − 0.224·7-s − 0.353·8-s + 1.46·9-s + 0.316·10-s − 1.55·11-s + 0.785·12-s + 0.119·13-s + 0.158·14-s − 0.702·15-s + 0.250·16-s + 0.861·17-s − 1.03·18-s − 1.17·19-s − 0.223·20-s − 0.353·21-s + 1.09·22-s + 1.40·23-s − 0.555·24-s + 0.200·25-s − 0.0844·26-s + 0.731·27-s − 0.112·28-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 \( 1 - 2.71T + 3T^{2} \)
7 \( 1 + 0.594T + 7T^{2} \)
11 \( 1 + 5.15T + 11T^{2} \)
13 \( 1 - 0.430T + 13T^{2} \)
17 \( 1 - 3.55T + 17T^{2} \)
19 \( 1 + 5.11T + 19T^{2} \)
23 \( 1 - 6.74T + 23T^{2} \)
29 \( 1 + 7.79T + 29T^{2} \)
31 \( 1 + 8.50T + 31T^{2} \)
37 \( 1 + 0.601T + 37T^{2} \)
41 \( 1 - 4.94T + 41T^{2} \)
43 \( 1 - 2.44T + 43T^{2} \)
47 \( 1 + 2.58T + 47T^{2} \)
53 \( 1 - 3.18T + 53T^{2} \)
59 \( 1 + 14.3T + 59T^{2} \)
61 \( 1 + 2.26T + 61T^{2} \)
67 \( 1 - 2.61T + 67T^{2} \)
71 \( 1 - 6.99T + 71T^{2} \)
73 \( 1 - 5.20T + 73T^{2} \)
79 \( 1 + 17.2T + 79T^{2} \)
83 \( 1 + 6.99T + 83T^{2} \)
89 \( 1 + 12.7T + 89T^{2} \)
97 \( 1 + 1.67T + 97T^{2} \)
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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

−8.164958856552195630426616709568, −7.48991331124478369861889427265, −7.21695097011808415436611224986, −5.95241224285457159444929295449, −5.03748729149150461248610488639, −3.90556398430994345227216529320, −3.16679218893763284403571255401, −2.54839929446008896674520973781, −1.61511181241831230246255718284, 0, 1.61511181241831230246255718284, 2.54839929446008896674520973781, 3.16679218893763284403571255401, 3.90556398430994345227216529320, 5.03748729149150461248610488639, 5.95241224285457159444929295449, 7.21695097011808415436611224986, 7.48991331124478369861889427265, 8.164958856552195630426616709568

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