Properties

Degree 2
Conductor $ 2 \cdot 4021 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3.12·3-s + 4-s + 1.55·5-s + 3.12·6-s + 0.219·7-s − 8-s + 6.79·9-s − 1.55·10-s − 3.38·11-s − 3.12·12-s − 1.24·13-s − 0.219·14-s − 4.86·15-s + 16-s − 6.07·17-s − 6.79·18-s − 4.37·19-s + 1.55·20-s − 0.687·21-s + 3.38·22-s − 0.273·23-s + 3.12·24-s − 2.58·25-s + 1.24·26-s − 11.8·27-s + 0.219·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.80·3-s + 0.5·4-s + 0.694·5-s + 1.27·6-s + 0.0830·7-s − 0.353·8-s + 2.26·9-s − 0.491·10-s − 1.01·11-s − 0.903·12-s − 0.344·13-s − 0.0587·14-s − 1.25·15-s + 0.250·16-s − 1.47·17-s − 1.60·18-s − 1.00·19-s + 0.347·20-s − 0.150·21-s + 0.721·22-s − 0.0570·23-s + 0.638·24-s − 0.517·25-s + 0.243·26-s − 2.28·27-s + 0.0415·28-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  =  0
Selberg data  =  $(2,\ 8042,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.05483821635$
$L(\frac12)$  $\approx$  $0.05483821635$
$L(\frac{3}{2})$   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 \( 1 + 3.12T + 3T^{2} \)
5 \( 1 - 1.55T + 5T^{2} \)
7 \( 1 - 0.219T + 7T^{2} \)
11 \( 1 + 3.38T + 11T^{2} \)
13 \( 1 + 1.24T + 13T^{2} \)
17 \( 1 + 6.07T + 17T^{2} \)
19 \( 1 + 4.37T + 19T^{2} \)
23 \( 1 + 0.273T + 23T^{2} \)
29 \( 1 + 9.14T + 29T^{2} \)
31 \( 1 + 0.0762T + 31T^{2} \)
37 \( 1 + 10.1T + 37T^{2} \)
41 \( 1 + 10.4T + 41T^{2} \)
43 \( 1 + 2.44T + 43T^{2} \)
47 \( 1 + 4.26T + 47T^{2} \)
53 \( 1 - 0.795T + 53T^{2} \)
59 \( 1 + 2.78T + 59T^{2} \)
61 \( 1 + 9.70T + 61T^{2} \)
67 \( 1 - 5.80T + 67T^{2} \)
71 \( 1 - 4.38T + 71T^{2} \)
73 \( 1 - 1.82T + 73T^{2} \)
79 \( 1 + 2.72T + 79T^{2} \)
83 \( 1 - 9.27T + 83T^{2} \)
89 \( 1 + 12.0T + 89T^{2} \)
97 \( 1 - 10.0T + 97T^{2} \)
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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

−7.66691208513724314794856384870, −6.97063602755726336151976925289, −6.42949336348929105740316436489, −5.88036052300639044094941518705, −5.11811337859762501232653243744, −4.72214048580434244182901098380, −3.57016813273614964822297926321, −2.09250303088936283907743093433, −1.74903638591219483228235751125, −0.13547659118675836025914444823, 0.13547659118675836025914444823, 1.74903638591219483228235751125, 2.09250303088936283907743093433, 3.57016813273614964822297926321, 4.72214048580434244182901098380, 5.11811337859762501232653243744, 5.88036052300639044094941518705, 6.42949336348929105740316436489, 6.97063602755726336151976925289, 7.66691208513724314794856384870

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