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  − 0.324·3-s − 5-s − 1.35·7-s − 2.89·9-s − 2.28·11-s − 1.54·13-s + 0.324·15-s + 7.82·17-s + 4.53·19-s + 0.438·21-s + 0.446·23-s + 25-s + 1.91·27-s + 2.54·29-s − 4.39·31-s + 0.740·33-s + 1.35·35-s − 5.52·37-s + 0.501·39-s + 2.55·41-s − 1.78·43-s + 2.89·45-s − 5.32·47-s − 5.16·49-s − 2.53·51-s + 0.0646·53-s + 2.28·55-s + ⋯
L(s)  = 1  − 0.187·3-s − 0.447·5-s − 0.511·7-s − 0.964·9-s − 0.688·11-s − 0.429·13-s + 0.0836·15-s + 1.89·17-s + 1.04·19-s + 0.0956·21-s + 0.0930·23-s + 0.200·25-s + 0.367·27-s + 0.473·29-s − 0.789·31-s + 0.128·33-s + 0.228·35-s − 0.908·37-s + 0.0803·39-s + 0.398·41-s − 0.272·43-s + 0.431·45-s − 0.776·47-s − 0.738·49-s − 0.355·51-s + 0.00887·53-s + 0.307·55-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 \( 1 + 0.324T + 3T^{2} \)
7 \( 1 + 1.35T + 7T^{2} \)
11 \( 1 + 2.28T + 11T^{2} \)
13 \( 1 + 1.54T + 13T^{2} \)
17 \( 1 - 7.82T + 17T^{2} \)
19 \( 1 - 4.53T + 19T^{2} \)
23 \( 1 - 0.446T + 23T^{2} \)
29 \( 1 - 2.54T + 29T^{2} \)
31 \( 1 + 4.39T + 31T^{2} \)
37 \( 1 + 5.52T + 37T^{2} \)
41 \( 1 - 2.55T + 41T^{2} \)
43 \( 1 + 1.78T + 43T^{2} \)
47 \( 1 + 5.32T + 47T^{2} \)
53 \( 1 - 0.0646T + 53T^{2} \)
59 \( 1 - 8.28T + 59T^{2} \)
61 \( 1 + 4.00T + 61T^{2} \)
67 \( 1 - 12.9T + 67T^{2} \)
71 \( 1 - 10.6T + 71T^{2} \)
73 \( 1 + 8.01T + 73T^{2} \)
79 \( 1 - 2.47T + 79T^{2} \)
83 \( 1 + 4.09T + 83T^{2} \)
89 \( 1 - 8.32T + 89T^{2} \)
97 \( 1 - 13.1T + 97T^{2} \)
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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

−7.61490899011694972223131209291, −6.85056735188474463571223633822, −6.01179662350083819912504149351, −5.28748555414138424350977001686, −4.98871468672910242057076605109, −3.56238611925007658525348105200, −3.28319358487144784977064605283, −2.40690447128626917001964840564, −1.05237591476129300001690471069, 0, 1.05237591476129300001690471069, 2.40690447128626917001964840564, 3.28319358487144784977064605283, 3.56238611925007658525348105200, 4.98871468672910242057076605109, 5.28748555414138424350977001686, 6.01179662350083819912504149351, 6.85056735188474463571223633822, 7.61490899011694972223131209291

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