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 − 2.10·3-s + 4-s − 2.86·5-s + 2.10·6-s + 3.42·7-s − 8-s + 1.42·9-s + 2.86·10-s − 1.51·11-s − 2.10·12-s − 1.17·13-s − 3.42·14-s + 6.02·15-s + 16-s − 7.28·17-s − 1.42·18-s − 3.37·19-s − 2.86·20-s − 7.21·21-s + 1.51·22-s − 2.46·23-s + 2.10·24-s + 3.21·25-s + 1.17·26-s + 3.31·27-s + 3.42·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.21·3-s + 0.5·4-s − 1.28·5-s + 0.858·6-s + 1.29·7-s − 0.353·8-s + 0.474·9-s + 0.906·10-s − 0.455·11-s − 0.607·12-s − 0.325·13-s − 0.916·14-s + 1.55·15-s + 0.250·16-s − 1.76·17-s − 0.335·18-s − 0.775·19-s − 0.640·20-s − 1.57·21-s + 0.322·22-s − 0.514·23-s + 0.429·24-s + 0.642·25-s + 0.230·26-s + 0.637·27-s + 0.648·28-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  =  $(2,\ 8042,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.06374228567$
$L(\frac12)$  $\approx$  $0.06374228567$
$L(\frac{3}{2})$   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 \( 1 + 2.10T + 3T^{2} \)
5 \( 1 + 2.86T + 5T^{2} \)
7 \( 1 - 3.42T + 7T^{2} \)
11 \( 1 + 1.51T + 11T^{2} \)
13 \( 1 + 1.17T + 13T^{2} \)
17 \( 1 + 7.28T + 17T^{2} \)
19 \( 1 + 3.37T + 19T^{2} \)
23 \( 1 + 2.46T + 23T^{2} \)
29 \( 1 - 4.84T + 29T^{2} \)
31 \( 1 + 8.53T + 31T^{2} \)
37 \( 1 + 3.32T + 37T^{2} \)
41 \( 1 + 9.68T + 41T^{2} \)
43 \( 1 + 1.96T + 43T^{2} \)
47 \( 1 + 0.645T + 47T^{2} \)
53 \( 1 + 9.65T + 53T^{2} \)
59 \( 1 + 0.822T + 59T^{2} \)
61 \( 1 - 10.3T + 61T^{2} \)
67 \( 1 + 5.13T + 67T^{2} \)
71 \( 1 + 10.7T + 71T^{2} \)
73 \( 1 - 6.12T + 73T^{2} \)
79 \( 1 - 15.3T + 79T^{2} \)
83 \( 1 + 6.42T + 83T^{2} \)
89 \( 1 + 12.7T + 89T^{2} \)
97 \( 1 - 3.11T + 97T^{2} \)
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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

−7.968083186902729872434241005424, −7.08081053015366510187882582270, −6.69864262611089532035689930644, −5.77194066612621251850474394793, −4.88874728251733576065268466823, −4.58672836047389331583732577300, −3.65253902300766238675551973112, −2.39353616750017564675215789895, −1.53942671186338691435202544743, −0.14823969002668146778152926996, 0.14823969002668146778152926996, 1.53942671186338691435202544743, 2.39353616750017564675215789895, 3.65253902300766238675551973112, 4.58672836047389331583732577300, 4.88874728251733576065268466823, 5.77194066612621251850474394793, 6.69864262611089532035689930644, 7.08081053015366510187882582270, 7.968083186902729872434241005424

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