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  − 0.750·3-s − 5-s − 4.31·7-s − 2.43·9-s + 1.26·11-s + 0.840·13-s + 0.750·15-s + 1.74·17-s + 4.53·19-s + 3.23·21-s − 7.00·23-s + 25-s + 4.07·27-s − 5.86·29-s + 5.85·31-s − 0.947·33-s + 4.31·35-s + 9.40·37-s − 0.630·39-s − 6.00·41-s + 6.28·43-s + 2.43·45-s + 1.12·47-s + 11.6·49-s − 1.30·51-s + 0.740·53-s − 1.26·55-s + ⋯
L(s)  = 1  − 0.433·3-s − 0.447·5-s − 1.63·7-s − 0.812·9-s + 0.380·11-s + 0.233·13-s + 0.193·15-s + 0.422·17-s + 1.03·19-s + 0.706·21-s − 1.46·23-s + 0.200·25-s + 0.784·27-s − 1.08·29-s + 1.05·31-s − 0.164·33-s + 0.729·35-s + 1.54·37-s − 0.100·39-s − 0.937·41-s + 0.958·43-s + 0.363·45-s + 0.163·47-s + 1.66·49-s − 0.183·51-s + 0.101·53-s − 0.170·55-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 \( 1 + 0.750T + 3T^{2} \)
7 \( 1 + 4.31T + 7T^{2} \)
11 \( 1 - 1.26T + 11T^{2} \)
13 \( 1 - 0.840T + 13T^{2} \)
17 \( 1 - 1.74T + 17T^{2} \)
19 \( 1 - 4.53T + 19T^{2} \)
23 \( 1 + 7.00T + 23T^{2} \)
29 \( 1 + 5.86T + 29T^{2} \)
31 \( 1 - 5.85T + 31T^{2} \)
37 \( 1 - 9.40T + 37T^{2} \)
41 \( 1 + 6.00T + 41T^{2} \)
43 \( 1 - 6.28T + 43T^{2} \)
47 \( 1 - 1.12T + 47T^{2} \)
53 \( 1 - 0.740T + 53T^{2} \)
59 \( 1 + 1.87T + 59T^{2} \)
61 \( 1 + 4.43T + 61T^{2} \)
67 \( 1 + 2.32T + 67T^{2} \)
71 \( 1 + 11.5T + 71T^{2} \)
73 \( 1 - 14.5T + 73T^{2} \)
79 \( 1 - 16.2T + 79T^{2} \)
83 \( 1 + 6.59T + 83T^{2} \)
89 \( 1 + 9.88T + 89T^{2} \)
97 \( 1 - 4.74T + 97T^{2} \)
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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

−7.48414072943001703246911133219, −6.65201583154067798944540203872, −6.01504288857951917354909488534, −5.72428899601038172802727901324, −4.63340429179166672014745363726, −3.71982123321061623275838473044, −3.25123948438750637451580664481, −2.42128268293822831676940906339, −0.939350364232261011605637152371, 0, 0.939350364232261011605637152371, 2.42128268293822831676940906339, 3.25123948438750637451580664481, 3.71982123321061623275838473044, 4.63340429179166672014745363726, 5.72428899601038172802727901324, 6.01504288857951917354909488534, 6.65201583154067798944540203872, 7.48414072943001703246911133219

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