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  + 2.01·3-s − 5-s + 1.12·7-s + 1.06·9-s + 1.62·11-s − 5.08·13-s − 2.01·15-s + 0.834·17-s − 2.00·19-s + 2.26·21-s − 2.97·23-s + 25-s − 3.90·27-s + 7.67·29-s + 9.19·31-s + 3.27·33-s − 1.12·35-s − 6.56·37-s − 10.2·39-s − 11.5·41-s − 2.96·43-s − 1.06·45-s − 11.2·47-s − 5.73·49-s + 1.68·51-s − 1.53·53-s − 1.62·55-s + ⋯
L(s)  = 1  + 1.16·3-s − 0.447·5-s + 0.424·7-s + 0.354·9-s + 0.489·11-s − 1.41·13-s − 0.520·15-s + 0.202·17-s − 0.458·19-s + 0.493·21-s − 0.620·23-s + 0.200·25-s − 0.750·27-s + 1.42·29-s + 1.65·31-s + 0.570·33-s − 0.189·35-s − 1.07·37-s − 1.64·39-s − 1.80·41-s − 0.451·43-s − 0.158·45-s − 1.64·47-s − 0.819·49-s + 0.235·51-s − 0.211·53-s − 0.219·55-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 \( 1 - 2.01T + 3T^{2} \)
7 \( 1 - 1.12T + 7T^{2} \)
11 \( 1 - 1.62T + 11T^{2} \)
13 \( 1 + 5.08T + 13T^{2} \)
17 \( 1 - 0.834T + 17T^{2} \)
19 \( 1 + 2.00T + 19T^{2} \)
23 \( 1 + 2.97T + 23T^{2} \)
29 \( 1 - 7.67T + 29T^{2} \)
31 \( 1 - 9.19T + 31T^{2} \)
37 \( 1 + 6.56T + 37T^{2} \)
41 \( 1 + 11.5T + 41T^{2} \)
43 \( 1 + 2.96T + 43T^{2} \)
47 \( 1 + 11.2T + 47T^{2} \)
53 \( 1 + 1.53T + 53T^{2} \)
59 \( 1 + 2.89T + 59T^{2} \)
61 \( 1 + 6.38T + 61T^{2} \)
67 \( 1 - 13.2T + 67T^{2} \)
71 \( 1 + 7.85T + 71T^{2} \)
73 \( 1 - 14.9T + 73T^{2} \)
79 \( 1 + 7.45T + 79T^{2} \)
83 \( 1 - 12.0T + 83T^{2} \)
89 \( 1 - 11.9T + 89T^{2} \)
97 \( 1 + 12.9T + 97T^{2} \)
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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

−7.74017989336909828989552751687, −6.84331148975867051567562055235, −6.38309181638511074419593579322, −5.04679672456844666669424653909, −4.71248476369471219529377970875, −3.74296966222128485937779082184, −3.07985040501476524712063202100, −2.34969140840042865614521976545, −1.50251394353462875842998383170, 0, 1.50251394353462875842998383170, 2.34969140840042865614521976545, 3.07985040501476524712063202100, 3.74296966222128485937779082184, 4.71248476369471219529377970875, 5.04679672456844666669424653909, 6.38309181638511074419593579322, 6.84331148975867051567562055235, 7.74017989336909828989552751687

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