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  + 0.167·3-s − 5-s − 3.89·7-s − 2.97·9-s + 3.80·11-s + 0.290·13-s − 0.167·15-s − 0.491·17-s − 4.25·19-s − 0.651·21-s + 8.71·23-s + 25-s − 1.00·27-s + 2.00·29-s + 2.46·31-s + 0.637·33-s + 3.89·35-s + 8.86·37-s + 0.0486·39-s − 6.82·41-s − 10.9·43-s + 2.97·45-s + 5.14·47-s + 8.14·49-s − 0.0823·51-s + 4.63·53-s − 3.80·55-s + ⋯
L(s)  = 1  + 0.0967·3-s − 0.447·5-s − 1.47·7-s − 0.990·9-s + 1.14·11-s + 0.0805·13-s − 0.0432·15-s − 0.119·17-s − 0.977·19-s − 0.142·21-s + 1.81·23-s + 0.200·25-s − 0.192·27-s + 0.372·29-s + 0.442·31-s + 0.111·33-s + 0.657·35-s + 1.45·37-s + 0.00779·39-s − 1.06·41-s − 1.66·43-s + 0.443·45-s + 0.750·47-s + 1.16·49-s − 0.0115·51-s + 0.636·53-s − 0.513·55-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 \( 1 - 0.167T + 3T^{2} \)
7 \( 1 + 3.89T + 7T^{2} \)
11 \( 1 - 3.80T + 11T^{2} \)
13 \( 1 - 0.290T + 13T^{2} \)
17 \( 1 + 0.491T + 17T^{2} \)
19 \( 1 + 4.25T + 19T^{2} \)
23 \( 1 - 8.71T + 23T^{2} \)
29 \( 1 - 2.00T + 29T^{2} \)
31 \( 1 - 2.46T + 31T^{2} \)
37 \( 1 - 8.86T + 37T^{2} \)
41 \( 1 + 6.82T + 41T^{2} \)
43 \( 1 + 10.9T + 43T^{2} \)
47 \( 1 - 5.14T + 47T^{2} \)
53 \( 1 - 4.63T + 53T^{2} \)
59 \( 1 - 11.0T + 59T^{2} \)
61 \( 1 + 0.286T + 61T^{2} \)
67 \( 1 + 3.22T + 67T^{2} \)
71 \( 1 + 2.53T + 71T^{2} \)
73 \( 1 + 5.56T + 73T^{2} \)
79 \( 1 + 10.6T + 79T^{2} \)
83 \( 1 - 1.47T + 83T^{2} \)
89 \( 1 - 8.34T + 89T^{2} \)
97 \( 1 + 10.4T + 97T^{2} \)
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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

−7.32372393849332964195200729710, −6.63315862642851261619391323395, −6.36596614249284170135013461336, −5.49110819016048079754548057052, −4.55404950268238134504614315730, −3.75752616334045722147833079629, −3.14270606000329653858764825175, −2.49849364161077311646920739425, −1.06822696942609025334577132040, 0, 1.06822696942609025334577132040, 2.49849364161077311646920739425, 3.14270606000329653858764825175, 3.75752616334045722147833079629, 4.55404950268238134504614315730, 5.49110819016048079754548057052, 6.36596614249284170135013461336, 6.63315862642851261619391323395, 7.32372393849332964195200729710

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