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  + 2.32·3-s − 5-s − 2.44·7-s + 2.40·9-s − 5.19·11-s + 2.45·13-s − 2.32·15-s + 3.30·17-s + 7.37·19-s − 5.69·21-s − 2.41·23-s + 25-s − 1.38·27-s − 0.883·29-s − 0.774·31-s − 12.0·33-s + 2.44·35-s − 3.11·37-s + 5.70·39-s + 11.0·41-s − 7.16·43-s − 2.40·45-s + 5.67·47-s − 1.00·49-s + 7.68·51-s − 10.4·53-s + 5.19·55-s + ⋯
L(s)  = 1  + 1.34·3-s − 0.447·5-s − 0.925·7-s + 0.801·9-s − 1.56·11-s + 0.681·13-s − 0.600·15-s + 0.801·17-s + 1.69·19-s − 1.24·21-s − 0.503·23-s + 0.200·25-s − 0.266·27-s − 0.163·29-s − 0.139·31-s − 2.10·33-s + 0.413·35-s − 0.512·37-s + 0.914·39-s + 1.73·41-s − 1.09·43-s − 0.358·45-s + 0.827·47-s − 0.143·49-s + 1.07·51-s − 1.43·53-s + 0.701·55-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 \( 1 - 2.32T + 3T^{2} \)
7 \( 1 + 2.44T + 7T^{2} \)
11 \( 1 + 5.19T + 11T^{2} \)
13 \( 1 - 2.45T + 13T^{2} \)
17 \( 1 - 3.30T + 17T^{2} \)
19 \( 1 - 7.37T + 19T^{2} \)
23 \( 1 + 2.41T + 23T^{2} \)
29 \( 1 + 0.883T + 29T^{2} \)
31 \( 1 + 0.774T + 31T^{2} \)
37 \( 1 + 3.11T + 37T^{2} \)
41 \( 1 - 11.0T + 41T^{2} \)
43 \( 1 + 7.16T + 43T^{2} \)
47 \( 1 - 5.67T + 47T^{2} \)
53 \( 1 + 10.4T + 53T^{2} \)
59 \( 1 + 8.65T + 59T^{2} \)
61 \( 1 + 11.8T + 61T^{2} \)
67 \( 1 - 7.03T + 67T^{2} \)
71 \( 1 + 7.87T + 71T^{2} \)
73 \( 1 - 8.30T + 73T^{2} \)
79 \( 1 + 12.0T + 79T^{2} \)
83 \( 1 + 4.17T + 83T^{2} \)
89 \( 1 + 9.39T + 89T^{2} \)
97 \( 1 + 16.7T + 97T^{2} \)
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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

−7.66401842269759020857920356137, −7.15190198024425639411216949175, −6.01755874742692749824692317526, −5.47985015365171362876247468970, −4.50238231741634672288939128037, −3.47745466639925396081563120565, −3.18726332929609965392041333046, −2.57520450038383011255849751028, −1.38635712212457435354054933021, 0, 1.38635712212457435354054933021, 2.57520450038383011255849751028, 3.18726332929609965392041333046, 3.47745466639925396081563120565, 4.50238231741634672288939128037, 5.47985015365171362876247468970, 6.01755874742692749824692317526, 7.15190198024425639411216949175, 7.66401842269759020857920356137

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