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 − 3.34·3-s + 4-s − 5-s + 3.34·6-s + 3.45·7-s − 8-s + 8.17·9-s + 10-s + 3.65·11-s − 3.34·12-s + 3.49·13-s − 3.45·14-s + 3.34·15-s + 16-s + 4.44·17-s − 8.17·18-s − 6.26·19-s − 20-s − 11.5·21-s − 3.65·22-s − 7.57·23-s + 3.34·24-s + 25-s − 3.49·26-s − 17.3·27-s + 3.45·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.93·3-s + 0.5·4-s − 0.447·5-s + 1.36·6-s + 1.30·7-s − 0.353·8-s + 2.72·9-s + 0.316·10-s + 1.10·11-s − 0.965·12-s + 0.969·13-s − 0.923·14-s + 0.863·15-s + 0.250·16-s + 1.07·17-s − 1.92·18-s − 1.43·19-s − 0.223·20-s − 2.51·21-s − 0.778·22-s − 1.57·23-s + 0.682·24-s + 0.200·25-s − 0.685·26-s − 3.32·27-s + 0.652·28-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 \( 1 + 3.34T + 3T^{2} \)
7 \( 1 - 3.45T + 7T^{2} \)
11 \( 1 - 3.65T + 11T^{2} \)
13 \( 1 - 3.49T + 13T^{2} \)
17 \( 1 - 4.44T + 17T^{2} \)
19 \( 1 + 6.26T + 19T^{2} \)
23 \( 1 + 7.57T + 23T^{2} \)
29 \( 1 + 2.62T + 29T^{2} \)
31 \( 1 - 3.28T + 31T^{2} \)
37 \( 1 + 7.45T + 37T^{2} \)
41 \( 1 - 4.62T + 41T^{2} \)
43 \( 1 + 9.40T + 43T^{2} \)
47 \( 1 + 4.02T + 47T^{2} \)
53 \( 1 - 4.07T + 53T^{2} \)
59 \( 1 - 4.53T + 59T^{2} \)
61 \( 1 + 11.7T + 61T^{2} \)
67 \( 1 + 2.55T + 67T^{2} \)
71 \( 1 - 13.5T + 71T^{2} \)
73 \( 1 + 12.2T + 73T^{2} \)
79 \( 1 + 11.7T + 79T^{2} \)
83 \( 1 + 9.38T + 83T^{2} \)
89 \( 1 - 1.01T + 89T^{2} \)
97 \( 1 - 3.76T + 97T^{2} \)
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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

−8.084783801095869153305750683117, −7.29195813849000031274733124420, −6.49618751816545812082614188360, −6.01486912482474603658000884897, −5.24310896960540576469312849687, −4.32912218050717585152269738624, −3.80746132761395163977909508331, −1.72523139780543888368510786592, −1.27589973157118854733046516114, 0, 1.27589973157118854733046516114, 1.72523139780543888368510786592, 3.80746132761395163977909508331, 4.32912218050717585152269738624, 5.24310896960540576469312849687, 6.01486912482474603658000884897, 6.49618751816545812082614188360, 7.29195813849000031274733124420, 8.084783801095869153305750683117

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