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 − 1.47·3-s + 4-s + 5-s + 1.47·6-s − 2.32·7-s − 8-s − 0.820·9-s − 10-s + 1.34·11-s − 1.47·12-s + 3.18·13-s + 2.32·14-s − 1.47·15-s + 16-s + 3.91·17-s + 0.820·18-s − 4.40·19-s + 20-s + 3.42·21-s − 1.34·22-s − 4.06·23-s + 1.47·24-s + 25-s − 3.18·26-s + 5.64·27-s − 2.32·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.852·3-s + 0.5·4-s + 0.447·5-s + 0.602·6-s − 0.877·7-s − 0.353·8-s − 0.273·9-s − 0.316·10-s + 0.405·11-s − 0.426·12-s + 0.883·13-s + 0.620·14-s − 0.381·15-s + 0.250·16-s + 0.950·17-s + 0.193·18-s − 1.01·19-s + 0.223·20-s + 0.748·21-s − 0.286·22-s − 0.847·23-s + 0.301·24-s + 0.200·25-s − 0.624·26-s + 1.08·27-s − 0.438·28-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 \( 1 + 1.47T + 3T^{2} \)
7 \( 1 + 2.32T + 7T^{2} \)
11 \( 1 - 1.34T + 11T^{2} \)
13 \( 1 - 3.18T + 13T^{2} \)
17 \( 1 - 3.91T + 17T^{2} \)
19 \( 1 + 4.40T + 19T^{2} \)
23 \( 1 + 4.06T + 23T^{2} \)
29 \( 1 + 8.96T + 29T^{2} \)
31 \( 1 - 1.15T + 31T^{2} \)
37 \( 1 - 0.657T + 37T^{2} \)
41 \( 1 - 7.76T + 41T^{2} \)
43 \( 1 - 5.75T + 43T^{2} \)
47 \( 1 + 7.10T + 47T^{2} \)
53 \( 1 - 14.3T + 53T^{2} \)
59 \( 1 + 2.03T + 59T^{2} \)
61 \( 1 - 1.33T + 61T^{2} \)
67 \( 1 + 1.04T + 67T^{2} \)
71 \( 1 - 3.41T + 71T^{2} \)
73 \( 1 + 7.58T + 73T^{2} \)
79 \( 1 + 8.46T + 79T^{2} \)
83 \( 1 + 3.04T + 83T^{2} \)
89 \( 1 + 1.14T + 89T^{2} \)
97 \( 1 - 13.4T + 97T^{2} \)
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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

−8.182739465555414739566569148664, −7.28248627824577299927325372400, −6.44800203273882545820699310892, −5.95932852399261621235367285258, −5.54564825269203541495685432034, −4.19099668426758454747983181104, −3.35314274589004306032885178822, −2.29497988778835994437268230440, −1.14497977190670319467270237010, 0, 1.14497977190670319467270237010, 2.29497988778835994437268230440, 3.35314274589004306032885178822, 4.19099668426758454747983181104, 5.54564825269203541495685432034, 5.95932852399261621235367285258, 6.44800203273882545820699310892, 7.28248627824577299927325372400, 8.182739465555414739566569148664

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