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 − 0.876·3-s + 4-s − 5-s + 0.876·6-s + 4.75·7-s − 8-s − 2.23·9-s + 10-s + 0.395·11-s − 0.876·12-s − 1.38·13-s − 4.75·14-s + 0.876·15-s + 16-s + 1.97·17-s + 2.23·18-s − 4.95·19-s − 20-s − 4.16·21-s − 0.395·22-s + 7.99·23-s + 0.876·24-s + 25-s + 1.38·26-s + 4.58·27-s + 4.75·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.505·3-s + 0.5·4-s − 0.447·5-s + 0.357·6-s + 1.79·7-s − 0.353·8-s − 0.744·9-s + 0.316·10-s + 0.119·11-s − 0.252·12-s − 0.385·13-s − 1.27·14-s + 0.226·15-s + 0.250·16-s + 0.479·17-s + 0.526·18-s − 1.13·19-s − 0.223·20-s − 0.909·21-s − 0.0843·22-s + 1.66·23-s + 0.178·24-s + 0.200·25-s + 0.272·26-s + 0.882·27-s + 0.899·28-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 \( 1 + 0.876T + 3T^{2} \)
7 \( 1 - 4.75T + 7T^{2} \)
11 \( 1 - 0.395T + 11T^{2} \)
13 \( 1 + 1.38T + 13T^{2} \)
17 \( 1 - 1.97T + 17T^{2} \)
19 \( 1 + 4.95T + 19T^{2} \)
23 \( 1 - 7.99T + 23T^{2} \)
29 \( 1 + 5.94T + 29T^{2} \)
31 \( 1 + 7.55T + 31T^{2} \)
37 \( 1 + 7.50T + 37T^{2} \)
41 \( 1 + 3.64T + 41T^{2} \)
43 \( 1 + 1.96T + 43T^{2} \)
47 \( 1 - 10.0T + 47T^{2} \)
53 \( 1 + 9.91T + 53T^{2} \)
59 \( 1 - 9.82T + 59T^{2} \)
61 \( 1 + 5.61T + 61T^{2} \)
67 \( 1 + 12.8T + 67T^{2} \)
71 \( 1 - 2.66T + 71T^{2} \)
73 \( 1 + 4.87T + 73T^{2} \)
79 \( 1 - 9.52T + 79T^{2} \)
83 \( 1 + 11.3T + 83T^{2} \)
89 \( 1 - 7.21T + 89T^{2} \)
97 \( 1 + 10.6T + 97T^{2} \)
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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

−8.175837552201961683712883923696, −7.41670184254555900190193349897, −6.89914210073620948550113882631, −5.70629468454502811707229460422, −5.21283191101860258451123468265, −4.43699695566273733044013454409, −3.32909929477339773725148402407, −2.18097414017264382917163612671, −1.30616904632550181407291907193, 0, 1.30616904632550181407291907193, 2.18097414017264382917163612671, 3.32909929477339773725148402407, 4.43699695566273733044013454409, 5.21283191101860258451123468265, 5.70629468454502811707229460422, 6.89914210073620948550113882631, 7.41670184254555900190193349897, 8.175837552201961683712883923696

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