Properties

Degree 2
Conductor $ 2 \cdot 5 \cdot 401 $
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-s + 4-s + 5-s − 6-s + 3·7-s + 8-s − 2·9-s + 10-s − 3·11-s − 12-s + 4·13-s + 3·14-s − 15-s + 16-s − 7·17-s − 2·18-s + 20-s − 3·21-s − 3·22-s − 6·23-s − 24-s + 25-s + 4·26-s + 5·27-s + 3·28-s + 10·29-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 1.13·7-s + 0.353·8-s − 2/3·9-s + 0.316·10-s − 0.904·11-s − 0.288·12-s + 1.10·13-s + 0.801·14-s − 0.258·15-s + 1/4·16-s − 1.69·17-s − 0.471·18-s + 0.223·20-s − 0.654·21-s − 0.639·22-s − 1.25·23-s − 0.204·24-s + 1/5·25-s + 0.784·26-s + 0.962·27-s + 0.566·28-s + 1.85·29-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4010\)    =    \(2 \cdot 5 \cdot 401\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{4010} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 4010,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.837318408$
$L(\frac12)$  $\approx$  $2.837318408$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5,\;401\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;5,\;401\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 - T \)
5 \( 1 - T \)
401 \( 1 - T \)
good3 \( 1 + T + p T^{2} \)
7 \( 1 - 3 T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 7 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 14 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 3 T + p T^{2} \)
71 \( 1 + 3 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 - 3 T + p T^{2} \)
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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

−8.366185869663910773564503815141, −7.78868583277833887437993467032, −6.61378644987028618357181095553, −6.10946402786816038690191989043, −5.51542178024399932948913909686, −4.64198156805021315941819752698, −4.23449851555026021414871716821, −2.77488811174550060171323193305, −2.20830561425569631317168057215, −0.905014131053380638466907402001, 0.905014131053380638466907402001, 2.20830561425569631317168057215, 2.77488811174550060171323193305, 4.23449851555026021414871716821, 4.64198156805021315941819752698, 5.51542178024399932948913909686, 6.10946402786816038690191989043, 6.61378644987028618357181095553, 7.78868583277833887437993467032, 8.366185869663910773564503815141

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