Properties

Degree 2
Conductor $ 2^{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  − 2.61·3-s − 5-s − 1.36·7-s + 3.82·9-s − 3.20·11-s − 0.991·13-s + 2.61·15-s − 6.53·17-s + 2.43·19-s + 3.57·21-s + 2.25·23-s + 25-s − 2.15·27-s − 4.41·29-s + 5.35·31-s + 8.38·33-s + 1.36·35-s + 11.4·37-s + 2.58·39-s − 4.67·41-s − 2.19·43-s − 3.82·45-s − 0.392·47-s − 5.12·49-s + 17.0·51-s + 9.29·53-s + 3.20·55-s + ⋯
L(s)  = 1  − 1.50·3-s − 0.447·5-s − 0.517·7-s + 1.27·9-s − 0.967·11-s − 0.274·13-s + 0.674·15-s − 1.58·17-s + 0.559·19-s + 0.779·21-s + 0.469·23-s + 0.200·25-s − 0.415·27-s − 0.819·29-s + 0.962·31-s + 1.45·33-s + 0.231·35-s + 1.87·37-s + 0.414·39-s − 0.729·41-s − 0.334·43-s − 0.570·45-s − 0.0572·47-s − 0.732·49-s + 2.39·51-s + 1.27·53-s + 0.432·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8020\)    =    \(2^{2} \cdot 5 \cdot 401\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{8020} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 8020,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 \( 1 \)
5 \( 1 + T \)
401 \( 1 - T \)
good3 \( 1 + 2.61T + 3T^{2} \)
7 \( 1 + 1.36T + 7T^{2} \)
11 \( 1 + 3.20T + 11T^{2} \)
13 \( 1 + 0.991T + 13T^{2} \)
17 \( 1 + 6.53T + 17T^{2} \)
19 \( 1 - 2.43T + 19T^{2} \)
23 \( 1 - 2.25T + 23T^{2} \)
29 \( 1 + 4.41T + 29T^{2} \)
31 \( 1 - 5.35T + 31T^{2} \)
37 \( 1 - 11.4T + 37T^{2} \)
41 \( 1 + 4.67T + 41T^{2} \)
43 \( 1 + 2.19T + 43T^{2} \)
47 \( 1 + 0.392T + 47T^{2} \)
53 \( 1 - 9.29T + 53T^{2} \)
59 \( 1 + 1.84T + 59T^{2} \)
61 \( 1 + 5.30T + 61T^{2} \)
67 \( 1 - 9.05T + 67T^{2} \)
71 \( 1 - 1.03T + 71T^{2} \)
73 \( 1 - 15.0T + 73T^{2} \)
79 \( 1 + 14.6T + 79T^{2} \)
83 \( 1 - 1.71T + 83T^{2} \)
89 \( 1 - 13.6T + 89T^{2} \)
97 \( 1 - 3.23T + 97T^{2} \)
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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

−7.32820545603881840688629317852, −6.63639468254084789643538924209, −6.18699190252945322754075930700, −5.33113879875867904316012588753, −4.82939715286985259539576710737, −4.16978874142575698185987600642, −3.10472207054966887790341363263, −2.22414599676383152876377242460, −0.814548852009464511332244847293, 0, 0.814548852009464511332244847293, 2.22414599676383152876377242460, 3.10472207054966887790341363263, 4.16978874142575698185987600642, 4.82939715286985259539576710737, 5.33113879875867904316012588753, 6.18699190252945322754075930700, 6.63639468254084789643538924209, 7.32820545603881840688629317852

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