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 − 2.60·3-s + 4-s − 0.616·5-s − 2.60·6-s − 3.04·7-s + 8-s + 3.79·9-s − 0.616·10-s − 0.708·11-s − 2.60·12-s + 4.52·13-s − 3.04·14-s + 1.60·15-s + 16-s − 7.53·17-s + 3.79·18-s + 1.15·19-s − 0.616·20-s + 7.93·21-s − 0.708·22-s + 6.01·23-s − 2.60·24-s − 4.61·25-s + 4.52·26-s − 2.06·27-s − 3.04·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.50·3-s + 0.5·4-s − 0.275·5-s − 1.06·6-s − 1.15·7-s + 0.353·8-s + 1.26·9-s − 0.195·10-s − 0.213·11-s − 0.752·12-s + 1.25·13-s − 0.813·14-s + 0.415·15-s + 0.250·16-s − 1.82·17-s + 0.894·18-s + 0.264·19-s − 0.137·20-s + 1.73·21-s − 0.151·22-s + 1.25·23-s − 0.532·24-s − 0.923·25-s + 0.886·26-s − 0.398·27-s − 0.575·28-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 \( 1 + 2.60T + 3T^{2} \)
5 \( 1 + 0.616T + 5T^{2} \)
7 \( 1 + 3.04T + 7T^{2} \)
11 \( 1 + 0.708T + 11T^{2} \)
13 \( 1 - 4.52T + 13T^{2} \)
17 \( 1 + 7.53T + 17T^{2} \)
19 \( 1 - 1.15T + 19T^{2} \)
23 \( 1 - 6.01T + 23T^{2} \)
29 \( 1 - 3.05T + 29T^{2} \)
31 \( 1 + 4.59T + 31T^{2} \)
37 \( 1 - 0.716T + 37T^{2} \)
41 \( 1 - 5.61T + 41T^{2} \)
43 \( 1 - 1.06T + 43T^{2} \)
47 \( 1 - 5.03T + 47T^{2} \)
53 \( 1 + 6.45T + 53T^{2} \)
59 \( 1 - 1.48T + 59T^{2} \)
61 \( 1 + 1.93T + 61T^{2} \)
67 \( 1 + 2.98T + 67T^{2} \)
71 \( 1 - 11.7T + 71T^{2} \)
73 \( 1 - 5.50T + 73T^{2} \)
79 \( 1 - 10.9T + 79T^{2} \)
83 \( 1 + 1.61T + 83T^{2} \)
89 \( 1 - 2.45T + 89T^{2} \)
97 \( 1 - 12.5T + 97T^{2} \)
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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

−7.02603116251637525036571057010, −6.55282009475909868982910694807, −6.14120579619048013028105106315, −5.47926733303746687426060417824, −4.74023164717287388720659988171, −4.03633618338505222045745875195, −3.34216001545315296925023329953, −2.32933463438924950126679845701, −1.02901697468313988989131593572, 0, 1.02901697468313988989131593572, 2.32933463438924950126679845701, 3.34216001545315296925023329953, 4.03633618338505222045745875195, 4.74023164717287388720659988171, 5.47926733303746687426060417824, 6.14120579619048013028105106315, 6.55282009475909868982910694807, 7.02603116251637525036571057010

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