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 + 2.30·3-s + 4-s + 5-s − 2.30·6-s − 5.12·7-s − 8-s + 2.32·9-s − 10-s − 1.82·11-s + 2.30·12-s + 2.86·13-s + 5.12·14-s + 2.30·15-s + 16-s + 5.18·17-s − 2.32·18-s − 1.72·19-s + 20-s − 11.8·21-s + 1.82·22-s − 0.231·23-s − 2.30·24-s + 25-s − 2.86·26-s − 1.55·27-s − 5.12·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.33·3-s + 0.5·4-s + 0.447·5-s − 0.942·6-s − 1.93·7-s − 0.353·8-s + 0.775·9-s − 0.316·10-s − 0.550·11-s + 0.666·12-s + 0.795·13-s + 1.36·14-s + 0.595·15-s + 0.250·16-s + 1.25·17-s − 0.548·18-s − 0.395·19-s + 0.223·20-s − 2.58·21-s + 0.389·22-s − 0.0483·23-s − 0.471·24-s + 0.200·25-s − 0.562·26-s − 0.299·27-s − 0.968·28-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 \( 1 - 2.30T + 3T^{2} \)
7 \( 1 + 5.12T + 7T^{2} \)
11 \( 1 + 1.82T + 11T^{2} \)
13 \( 1 - 2.86T + 13T^{2} \)
17 \( 1 - 5.18T + 17T^{2} \)
19 \( 1 + 1.72T + 19T^{2} \)
23 \( 1 + 0.231T + 23T^{2} \)
29 \( 1 - 0.413T + 29T^{2} \)
31 \( 1 + 5.80T + 31T^{2} \)
37 \( 1 + 0.773T + 37T^{2} \)
41 \( 1 + 8.47T + 41T^{2} \)
43 \( 1 + 10.9T + 43T^{2} \)
47 \( 1 + 13.3T + 47T^{2} \)
53 \( 1 - 3.57T + 53T^{2} \)
59 \( 1 + 8.35T + 59T^{2} \)
61 \( 1 - 7.09T + 61T^{2} \)
67 \( 1 - 10.1T + 67T^{2} \)
71 \( 1 + 7.02T + 71T^{2} \)
73 \( 1 + 7.84T + 73T^{2} \)
79 \( 1 - 8.27T + 79T^{2} \)
83 \( 1 - 2.73T + 83T^{2} \)
89 \( 1 - 1.59T + 89T^{2} \)
97 \( 1 + 11.7T + 97T^{2} \)
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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

−8.319021853054874206447632820104, −7.50047079412507067232580469808, −6.72757513278983424528819927734, −6.13208626723513572867987527079, −5.26614695887212159545415139238, −3.54717049887883054460871976840, −3.40905533675463278514888034766, −2.55975168377336365124557300766, −1.56026130152732342930161177109, 0, 1.56026130152732342930161177109, 2.55975168377336365124557300766, 3.40905533675463278514888034766, 3.54717049887883054460871976840, 5.26614695887212159545415139238, 6.13208626723513572867987527079, 6.72757513278983424528819927734, 7.50047079412507067232580469808, 8.319021853054874206447632820104

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