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  − 2.59·3-s − 5-s + 1.42·7-s + 3.73·9-s + 3.63·11-s − 2.47·13-s + 2.59·15-s + 4.00·17-s + 0.503·19-s − 3.69·21-s − 8.01·23-s + 25-s − 1.90·27-s − 1.71·29-s − 6.28·31-s − 9.44·33-s − 1.42·35-s + 5.09·37-s + 6.43·39-s + 1.37·41-s + 7.23·43-s − 3.73·45-s + 7.49·47-s − 4.97·49-s − 10.4·51-s − 7.78·53-s − 3.63·55-s + ⋯
L(s)  = 1  − 1.49·3-s − 0.447·5-s + 0.538·7-s + 1.24·9-s + 1.09·11-s − 0.687·13-s + 0.670·15-s + 0.972·17-s + 0.115·19-s − 0.806·21-s − 1.67·23-s + 0.200·25-s − 0.366·27-s − 0.319·29-s − 1.12·31-s − 1.64·33-s − 0.240·35-s + 0.837·37-s + 1.03·39-s + 0.215·41-s + 1.10·43-s − 0.556·45-s + 1.09·47-s − 0.710·49-s − 1.45·51-s − 1.06·53-s − 0.490·55-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 \( 1 + 2.59T + 3T^{2} \)
7 \( 1 - 1.42T + 7T^{2} \)
11 \( 1 - 3.63T + 11T^{2} \)
13 \( 1 + 2.47T + 13T^{2} \)
17 \( 1 - 4.00T + 17T^{2} \)
19 \( 1 - 0.503T + 19T^{2} \)
23 \( 1 + 8.01T + 23T^{2} \)
29 \( 1 + 1.71T + 29T^{2} \)
31 \( 1 + 6.28T + 31T^{2} \)
37 \( 1 - 5.09T + 37T^{2} \)
41 \( 1 - 1.37T + 41T^{2} \)
43 \( 1 - 7.23T + 43T^{2} \)
47 \( 1 - 7.49T + 47T^{2} \)
53 \( 1 + 7.78T + 53T^{2} \)
59 \( 1 + 5.83T + 59T^{2} \)
61 \( 1 + 9.44T + 61T^{2} \)
67 \( 1 - 9.06T + 67T^{2} \)
71 \( 1 + 9.23T + 71T^{2} \)
73 \( 1 + 7.53T + 73T^{2} \)
79 \( 1 + 10.3T + 79T^{2} \)
83 \( 1 - 6.25T + 83T^{2} \)
89 \( 1 - 2.74T + 89T^{2} \)
97 \( 1 - 9.29T + 97T^{2} \)
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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

−7.55255836113861583222372009548, −6.68884001320374139198300000256, −5.91174289247972178876707826411, −5.61781127348984626837993889610, −4.60061973853961601958238388870, −4.23412610501610226345759248228, −3.28113033753927610587314548792, −1.93817683163853185849388935253, −1.06075026593322353374370857803, 0, 1.06075026593322353374370857803, 1.93817683163853185849388935253, 3.28113033753927610587314548792, 4.23412610501610226345759248228, 4.60061973853961601958238388870, 5.61781127348984626837993889610, 5.91174289247972178876707826411, 6.68884001320374139198300000256, 7.55255836113861583222372009548

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