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 − 0.918·3-s + 4-s − 5-s + 0.918·6-s − 3.92·7-s − 8-s − 2.15·9-s + 10-s + 1.54·11-s − 0.918·12-s − 0.562·13-s + 3.92·14-s + 0.918·15-s + 16-s − 6.59·17-s + 2.15·18-s + 6.11·19-s − 20-s + 3.60·21-s − 1.54·22-s + 4.55·23-s + 0.918·24-s + 25-s + 0.562·26-s + 4.73·27-s − 3.92·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.530·3-s + 0.5·4-s − 0.447·5-s + 0.375·6-s − 1.48·7-s − 0.353·8-s − 0.718·9-s + 0.316·10-s + 0.465·11-s − 0.265·12-s − 0.155·13-s + 1.04·14-s + 0.237·15-s + 0.250·16-s − 1.60·17-s + 0.508·18-s + 1.40·19-s − 0.223·20-s + 0.787·21-s − 0.329·22-s + 0.949·23-s + 0.187·24-s + 0.200·25-s + 0.110·26-s + 0.911·27-s − 0.741·28-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 \( 1 + 0.918T + 3T^{2} \)
7 \( 1 + 3.92T + 7T^{2} \)
11 \( 1 - 1.54T + 11T^{2} \)
13 \( 1 + 0.562T + 13T^{2} \)
17 \( 1 + 6.59T + 17T^{2} \)
19 \( 1 - 6.11T + 19T^{2} \)
23 \( 1 - 4.55T + 23T^{2} \)
29 \( 1 - 4.69T + 29T^{2} \)
31 \( 1 - 5.10T + 31T^{2} \)
37 \( 1 + 2.86T + 37T^{2} \)
41 \( 1 - 6.29T + 41T^{2} \)
43 \( 1 + 3.87T + 43T^{2} \)
47 \( 1 + 2.99T + 47T^{2} \)
53 \( 1 - 8.81T + 53T^{2} \)
59 \( 1 - 8.63T + 59T^{2} \)
61 \( 1 + 5.01T + 61T^{2} \)
67 \( 1 - 1.74T + 67T^{2} \)
71 \( 1 + 7.67T + 71T^{2} \)
73 \( 1 - 3.93T + 73T^{2} \)
79 \( 1 + 14.4T + 79T^{2} \)
83 \( 1 - 4.99T + 83T^{2} \)
89 \( 1 + 14.6T + 89T^{2} \)
97 \( 1 + 4.83T + 97T^{2} \)
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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

−8.258205930076386872130956675374, −7.08044233719543860585794011897, −6.78910715491639302825140078340, −6.08969005095913684494101883601, −5.24137352388360780168128785949, −4.22123687030090551656072482364, −3.14567038879806148831207899276, −2.64241627226355832816110911206, −0.971018027044129258548824435129, 0, 0.971018027044129258548824435129, 2.64241627226355832816110911206, 3.14567038879806148831207899276, 4.22123687030090551656072482364, 5.24137352388360780168128785949, 6.08969005095913684494101883601, 6.78910715491639302825140078340, 7.08044233719543860585794011897, 8.258205930076386872130956675374

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