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  + 2.66·3-s − 5-s + 1.97·7-s + 4.08·9-s − 1.73·11-s − 1.28·13-s − 2.66·15-s − 5.75·17-s + 5.34·19-s + 5.25·21-s − 8.91·23-s + 25-s + 2.88·27-s − 9.63·29-s − 2.06·31-s − 4.62·33-s − 1.97·35-s − 1.44·37-s − 3.43·39-s + 1.37·41-s + 2.31·43-s − 4.08·45-s − 6.46·47-s − 3.10·49-s − 15.3·51-s − 2.84·53-s + 1.73·55-s + ⋯
L(s)  = 1  + 1.53·3-s − 0.447·5-s + 0.745·7-s + 1.36·9-s − 0.523·11-s − 0.357·13-s − 0.687·15-s − 1.39·17-s + 1.22·19-s + 1.14·21-s − 1.85·23-s + 0.200·25-s + 0.556·27-s − 1.78·29-s − 0.371·31-s − 0.804·33-s − 0.333·35-s − 0.238·37-s − 0.549·39-s + 0.214·41-s + 0.352·43-s − 0.609·45-s − 0.943·47-s − 0.443·49-s − 2.14·51-s − 0.391·53-s + 0.234·55-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 \( 1 - 2.66T + 3T^{2} \)
7 \( 1 - 1.97T + 7T^{2} \)
11 \( 1 + 1.73T + 11T^{2} \)
13 \( 1 + 1.28T + 13T^{2} \)
17 \( 1 + 5.75T + 17T^{2} \)
19 \( 1 - 5.34T + 19T^{2} \)
23 \( 1 + 8.91T + 23T^{2} \)
29 \( 1 + 9.63T + 29T^{2} \)
31 \( 1 + 2.06T + 31T^{2} \)
37 \( 1 + 1.44T + 37T^{2} \)
41 \( 1 - 1.37T + 41T^{2} \)
43 \( 1 - 2.31T + 43T^{2} \)
47 \( 1 + 6.46T + 47T^{2} \)
53 \( 1 + 2.84T + 53T^{2} \)
59 \( 1 + 0.511T + 59T^{2} \)
61 \( 1 + 0.496T + 61T^{2} \)
67 \( 1 - 0.713T + 67T^{2} \)
71 \( 1 + 16.7T + 71T^{2} \)
73 \( 1 + 5.39T + 73T^{2} \)
79 \( 1 - 16.8T + 79T^{2} \)
83 \( 1 + 3.22T + 83T^{2} \)
89 \( 1 - 3.45T + 89T^{2} \)
97 \( 1 - 10.7T + 97T^{2} \)
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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

−7.67425077697401415130994680467, −7.23557878264059265782611954507, −6.14964156879394283171569304745, −5.23148628467560247183750679026, −4.46024737720731844485214691239, −3.80920597012635408840382036945, −3.10714914498094898357431973674, −2.18840371491872878602317483153, −1.71479755427052812727900853673, 0, 1.71479755427052812727900853673, 2.18840371491872878602317483153, 3.10714914498094898357431973674, 3.80920597012635408840382036945, 4.46024737720731844485214691239, 5.23148628467560247183750679026, 6.14964156879394283171569304745, 7.23557878264059265782611954507, 7.67425077697401415130994680467

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