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 − 3.05·3-s + 4-s − 1.41·5-s − 3.05·6-s − 1.76·7-s + 8-s + 6.30·9-s − 1.41·10-s + 0.990·11-s − 3.05·12-s + 2.81·13-s − 1.76·14-s + 4.30·15-s + 16-s − 1.33·17-s + 6.30·18-s − 3.59·19-s − 1.41·20-s + 5.39·21-s + 0.990·22-s − 1.70·23-s − 3.05·24-s − 3.00·25-s + 2.81·26-s − 10.0·27-s − 1.76·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.76·3-s + 0.5·4-s − 0.631·5-s − 1.24·6-s − 0.668·7-s + 0.353·8-s + 2.10·9-s − 0.446·10-s + 0.298·11-s − 0.880·12-s + 0.782·13-s − 0.472·14-s + 1.11·15-s + 0.250·16-s − 0.324·17-s + 1.48·18-s − 0.823·19-s − 0.315·20-s + 1.17·21-s + 0.211·22-s − 0.356·23-s − 0.622·24-s − 0.601·25-s + 0.553·26-s − 1.94·27-s − 0.334·28-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 \( 1 + 3.05T + 3T^{2} \)
5 \( 1 + 1.41T + 5T^{2} \)
7 \( 1 + 1.76T + 7T^{2} \)
11 \( 1 - 0.990T + 11T^{2} \)
13 \( 1 - 2.81T + 13T^{2} \)
17 \( 1 + 1.33T + 17T^{2} \)
19 \( 1 + 3.59T + 19T^{2} \)
23 \( 1 + 1.70T + 23T^{2} \)
29 \( 1 + 7.08T + 29T^{2} \)
31 \( 1 - 8.88T + 31T^{2} \)
37 \( 1 + 3.20T + 37T^{2} \)
41 \( 1 - 8.34T + 41T^{2} \)
43 \( 1 - 4.65T + 43T^{2} \)
47 \( 1 - 6.28T + 47T^{2} \)
53 \( 1 - 3.23T + 53T^{2} \)
59 \( 1 + 6.97T + 59T^{2} \)
61 \( 1 - 3.30T + 61T^{2} \)
67 \( 1 - 7.59T + 67T^{2} \)
71 \( 1 - 2.97T + 71T^{2} \)
73 \( 1 + 7.64T + 73T^{2} \)
79 \( 1 - 1.60T + 79T^{2} \)
83 \( 1 + 3.29T + 83T^{2} \)
89 \( 1 + 9.31T + 89T^{2} \)
97 \( 1 + 10.1T + 97T^{2} \)
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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

−7.16911109029858813317485046856, −6.51513199342774810335021514828, −6.02168762967167636817645526852, −5.61072099454211493218901872776, −4.55343241225603578814332583840, −4.16386206071107706157762533096, −3.47528056881948859918033908073, −2.19208235246353720467831687835, −1.02097698646952223122152209874, 0, 1.02097698646952223122152209874, 2.19208235246353720467831687835, 3.47528056881948859918033908073, 4.16386206071107706157762533096, 4.55343241225603578814332583840, 5.61072099454211493218901872776, 6.02168762967167636817645526852, 6.51513199342774810335021514828, 7.16911109029858813317485046856

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