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 − 3.13·3-s + 4-s + 5-s + 3.13·6-s + 2.49·7-s − 8-s + 6.79·9-s − 10-s − 1.60·11-s − 3.13·12-s − 0.554·13-s − 2.49·14-s − 3.13·15-s + 16-s + 5.26·17-s − 6.79·18-s − 2.32·19-s + 20-s − 7.82·21-s + 1.60·22-s + 4.26·23-s + 3.13·24-s + 25-s + 0.554·26-s − 11.8·27-s + 2.49·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.80·3-s + 0.5·4-s + 0.447·5-s + 1.27·6-s + 0.944·7-s − 0.353·8-s + 2.26·9-s − 0.316·10-s − 0.483·11-s − 0.903·12-s − 0.153·13-s − 0.667·14-s − 0.808·15-s + 0.250·16-s + 1.27·17-s − 1.60·18-s − 0.533·19-s + 0.223·20-s − 1.70·21-s + 0.341·22-s + 0.889·23-s + 0.638·24-s + 0.200·25-s + 0.108·26-s − 2.28·27-s + 0.472·28-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 \( 1 + 3.13T + 3T^{2} \)
7 \( 1 - 2.49T + 7T^{2} \)
11 \( 1 + 1.60T + 11T^{2} \)
13 \( 1 + 0.554T + 13T^{2} \)
17 \( 1 - 5.26T + 17T^{2} \)
19 \( 1 + 2.32T + 19T^{2} \)
23 \( 1 - 4.26T + 23T^{2} \)
29 \( 1 + 2.40T + 29T^{2} \)
31 \( 1 + 10.2T + 31T^{2} \)
37 \( 1 + 0.359T + 37T^{2} \)
41 \( 1 + 2.50T + 41T^{2} \)
43 \( 1 - 1.01T + 43T^{2} \)
47 \( 1 + 6.02T + 47T^{2} \)
53 \( 1 + 11.4T + 53T^{2} \)
59 \( 1 + 5.94T + 59T^{2} \)
61 \( 1 - 0.692T + 61T^{2} \)
67 \( 1 + 3.85T + 67T^{2} \)
71 \( 1 + 6.49T + 71T^{2} \)
73 \( 1 - 5.75T + 73T^{2} \)
79 \( 1 - 1.29T + 79T^{2} \)
83 \( 1 - 2.60T + 83T^{2} \)
89 \( 1 - 9.62T + 89T^{2} \)
97 \( 1 - 7.34T + 97T^{2} \)
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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

−7.81596506617252227213903634703, −7.42215725632832677856535560128, −6.53323599808328707839647432965, −5.85353182001760581489939407335, −5.18921943059932567154571545393, −4.74815286744035664864617721149, −3.42097093133713388531807582139, −1.91963133058607659587934320137, −1.22402491260016228744459577363, 0, 1.22402491260016228744459577363, 1.91963133058607659587934320137, 3.42097093133713388531807582139, 4.74815286744035664864617721149, 5.18921943059932567154571545393, 5.85353182001760581489939407335, 6.53323599808328707839647432965, 7.42215725632832677856535560128, 7.81596506617252227213903634703

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