Properties

Degree 2
Conductor $ 5 \cdot 11 \cdot 73 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 1.44·2-s + 0.0812·3-s + 0.0869·4-s + 5-s + 0.117·6-s + 1.22·7-s − 2.76·8-s − 2.99·9-s + 1.44·10-s + 11-s + 0.00707·12-s + 4.04·13-s + 1.77·14-s + 0.0812·15-s − 4.16·16-s − 0.951·17-s − 4.32·18-s + 6.58·19-s + 0.0869·20-s + 0.0998·21-s + 1.44·22-s + 7.77·23-s − 0.224·24-s + 25-s + 5.84·26-s − 0.487·27-s + 0.106·28-s + ⋯
L(s)  = 1  + 1.02·2-s + 0.0469·3-s + 0.0434·4-s + 0.447·5-s + 0.0479·6-s + 0.464·7-s − 0.977·8-s − 0.997·9-s + 0.456·10-s + 0.301·11-s + 0.00204·12-s + 1.12·13-s + 0.474·14-s + 0.0209·15-s − 1.04·16-s − 0.230·17-s − 1.01·18-s + 1.51·19-s + 0.0194·20-s + 0.0217·21-s + 0.307·22-s + 1.62·23-s − 0.0458·24-s + 0.200·25-s + 1.14·26-s − 0.0937·27-s + 0.0201·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4015\)    =    \(5 \cdot 11 \cdot 73\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{4015} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 4015,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.319023598$
$L(\frac12)$  $\approx$  $3.319023598$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{5,\;11,\;73\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{5,\;11,\;73\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad5 \( 1 - T \)
11 \( 1 - T \)
73 \( 1 - T \)
good2 \( 1 - 1.44T + 2T^{2} \)
3 \( 1 - 0.0812T + 3T^{2} \)
7 \( 1 - 1.22T + 7T^{2} \)
13 \( 1 - 4.04T + 13T^{2} \)
17 \( 1 + 0.951T + 17T^{2} \)
19 \( 1 - 6.58T + 19T^{2} \)
23 \( 1 - 7.77T + 23T^{2} \)
29 \( 1 + 6.21T + 29T^{2} \)
31 \( 1 + 5.11T + 31T^{2} \)
37 \( 1 + 4.08T + 37T^{2} \)
41 \( 1 + 1.52T + 41T^{2} \)
43 \( 1 - 8.93T + 43T^{2} \)
47 \( 1 - 7.83T + 47T^{2} \)
53 \( 1 + 10.1T + 53T^{2} \)
59 \( 1 - 12.5T + 59T^{2} \)
61 \( 1 - 4.37T + 61T^{2} \)
67 \( 1 - 15.3T + 67T^{2} \)
71 \( 1 + 4.94T + 71T^{2} \)
79 \( 1 + 3.79T + 79T^{2} \)
83 \( 1 - 17.4T + 83T^{2} \)
89 \( 1 - 8.86T + 89T^{2} \)
97 \( 1 - 3.39T + 97T^{2} \)
show more
show less
\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.668515806384133854492630021516, −7.63670365840780467517305980974, −6.76551208693445561099929721003, −5.91136213178498064034920813162, −5.41135547923721995778217628421, −4.87931064953679215686206558758, −3.68382868323843799547919601507, −3.29712419443327978727711504922, −2.22170369471682595244400457324, −0.920147660966838323207034042381, 0.920147660966838323207034042381, 2.22170369471682595244400457324, 3.29712419443327978727711504922, 3.68382868323843799547919601507, 4.87931064953679215686206558758, 5.41135547923721995778217628421, 5.91136213178498064034920813162, 6.76551208693445561099929721003, 7.63670365840780467517305980974, 8.668515806384133854492630021516

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