Properties

Degree 2
Conductor $ 2 \cdot 5 \cdot 13 \cdot 31 $
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 + 4-s + 5-s − 8-s − 3·9-s − 10-s − 2·11-s + 13-s + 16-s + 4·17-s + 3·18-s + 4·19-s + 20-s + 2·22-s − 2·23-s + 25-s − 26-s + 31-s − 32-s − 4·34-s − 3·36-s + 2·37-s − 4·38-s − 40-s + 6·41-s − 8·43-s − 2·44-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.353·8-s − 9-s − 0.316·10-s − 0.603·11-s + 0.277·13-s + 1/4·16-s + 0.970·17-s + 0.707·18-s + 0.917·19-s + 0.223·20-s + 0.426·22-s − 0.417·23-s + 1/5·25-s − 0.196·26-s + 0.179·31-s − 0.176·32-s − 0.685·34-s − 1/2·36-s + 0.328·37-s − 0.648·38-s − 0.158·40-s + 0.937·41-s − 1.21·43-s − 0.301·44-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4030\)    =    \(2 \cdot 5 \cdot 13 \cdot 31\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{4030} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 4030,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.259732793$
$L(\frac12)$  $\approx$  $1.259732793$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5,\;13,\;31\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;5,\;13,\;31\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + T \)
5 \( 1 - T \)
13 \( 1 - T \)
31 \( 1 - T \)
good3 \( 1 + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−8.337296170408014786469319928346, −7.951543552157607870170098822775, −7.09573548187363039508556929996, −6.19949040843258067314687506862, −5.60512336553810425499158447926, −4.93624793047737400875329064788, −3.51391561933045168855363528944, −2.87247694444336122333705469335, −1.89144565464029981894664761017, −0.70860755181105993924642892982, 0.70860755181105993924642892982, 1.89144565464029981894664761017, 2.87247694444336122333705469335, 3.51391561933045168855363528944, 4.93624793047737400875329064788, 5.60512336553810425499158447926, 6.19949040843258067314687506862, 7.09573548187363039508556929996, 7.951543552157607870170098822775, 8.337296170408014786469319928346

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