Properties

Degree 2
Conductor $ 2 \cdot 4021 $
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-s − 1.67·3-s + 4-s − 3.27·5-s − 1.67·6-s + 1.23·7-s + 8-s − 0.192·9-s − 3.27·10-s + 1.03·11-s − 1.67·12-s − 5.77·13-s + 1.23·14-s + 5.48·15-s + 16-s − 1.01·17-s − 0.192·18-s + 5.51·19-s − 3.27·20-s − 2.06·21-s + 1.03·22-s − 2.52·23-s − 1.67·24-s + 5.69·25-s − 5.77·26-s + 5.34·27-s + 1.23·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.967·3-s + 0.5·4-s − 1.46·5-s − 0.684·6-s + 0.464·7-s + 0.353·8-s − 0.0642·9-s − 1.03·10-s + 0.310·11-s − 0.483·12-s − 1.60·13-s + 0.328·14-s + 1.41·15-s + 0.250·16-s − 0.246·17-s − 0.0454·18-s + 1.26·19-s − 0.731·20-s − 0.449·21-s + 0.219·22-s − 0.526·23-s − 0.342·24-s + 1.13·25-s − 1.13·26-s + 1.02·27-s + 0.232·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8042\)    =    \(2 \cdot 4021\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{8042} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 8042,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;4021\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;4021\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 - T \)
4021 \( 1+O(T) \)
good3 \( 1 + 1.67T + 3T^{2} \)
5 \( 1 + 3.27T + 5T^{2} \)
7 \( 1 - 1.23T + 7T^{2} \)
11 \( 1 - 1.03T + 11T^{2} \)
13 \( 1 + 5.77T + 13T^{2} \)
17 \( 1 + 1.01T + 17T^{2} \)
19 \( 1 - 5.51T + 19T^{2} \)
23 \( 1 + 2.52T + 23T^{2} \)
29 \( 1 - 5.31T + 29T^{2} \)
31 \( 1 - 1.59T + 31T^{2} \)
37 \( 1 + 4.35T + 37T^{2} \)
41 \( 1 - 9.80T + 41T^{2} \)
43 \( 1 - 4.54T + 43T^{2} \)
47 \( 1 + 3.35T + 47T^{2} \)
53 \( 1 - 1.72T + 53T^{2} \)
59 \( 1 - 14.0T + 59T^{2} \)
61 \( 1 + 2.49T + 61T^{2} \)
67 \( 1 + 3.50T + 67T^{2} \)
71 \( 1 + 8.10T + 71T^{2} \)
73 \( 1 - 14.1T + 73T^{2} \)
79 \( 1 + 15.9T + 79T^{2} \)
83 \( 1 + 17.4T + 83T^{2} \)
89 \( 1 - 4.46T + 89T^{2} \)
97 \( 1 + 5.30T + 97T^{2} \)
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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

−7.30280839149442255910052487406, −6.88996466715091731808045179170, −5.94078900488733623500155615699, −5.25431673830830852683436555894, −4.66203893337640620301544654542, −4.18736719034373162104426860821, −3.21599654456242455563531707358, −2.46996126326828881499715023968, −1.04039811715934283077477717307, 0, 1.04039811715934283077477717307, 2.46996126326828881499715023968, 3.21599654456242455563531707358, 4.18736719034373162104426860821, 4.66203893337640620301544654542, 5.25431673830830852683436555894, 5.94078900488733623500155615699, 6.88996466715091731808045179170, 7.30280839149442255910052487406

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