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  − 1.93·2-s − 1.55·3-s + 1.73·4-s + 5-s + 2.99·6-s + 5.03·7-s + 0.518·8-s − 0.596·9-s − 1.93·10-s + 11-s − 2.68·12-s + 4.93·13-s − 9.73·14-s − 1.55·15-s − 4.46·16-s − 1.88·17-s + 1.15·18-s − 5.80·19-s + 1.73·20-s − 7.81·21-s − 1.93·22-s + 2.09·23-s − 0.803·24-s + 25-s − 9.53·26-s + 5.57·27-s + 8.72·28-s + ⋯
L(s)  = 1  − 1.36·2-s − 0.895·3-s + 0.865·4-s + 0.447·5-s + 1.22·6-s + 1.90·7-s + 0.183·8-s − 0.198·9-s − 0.610·10-s + 0.301·11-s − 0.774·12-s + 1.36·13-s − 2.60·14-s − 0.400·15-s − 1.11·16-s − 0.456·17-s + 0.271·18-s − 1.33·19-s + 0.387·20-s − 1.70·21-s − 0.411·22-s + 0.436·23-s − 0.164·24-s + 0.200·25-s − 1.86·26-s + 1.07·27-s + 1.64·28-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  =  $(2,\ 4015,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.9617938143$
$L(\frac12)$  $\approx$  $0.9617938143$
$L(\frac{3}{2})$   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 \( 1 + 1.93T + 2T^{2} \)
3 \( 1 + 1.55T + 3T^{2} \)
7 \( 1 - 5.03T + 7T^{2} \)
13 \( 1 - 4.93T + 13T^{2} \)
17 \( 1 + 1.88T + 17T^{2} \)
19 \( 1 + 5.80T + 19T^{2} \)
23 \( 1 - 2.09T + 23T^{2} \)
29 \( 1 - 7.08T + 29T^{2} \)
31 \( 1 - 0.155T + 31T^{2} \)
37 \( 1 + 2.57T + 37T^{2} \)
41 \( 1 - 3.66T + 41T^{2} \)
43 \( 1 + 6.60T + 43T^{2} \)
47 \( 1 - 0.494T + 47T^{2} \)
53 \( 1 + 0.831T + 53T^{2} \)
59 \( 1 - 7.98T + 59T^{2} \)
61 \( 1 - 12.2T + 61T^{2} \)
67 \( 1 - 4.93T + 67T^{2} \)
71 \( 1 - 8.14T + 71T^{2} \)
79 \( 1 + 10.7T + 79T^{2} \)
83 \( 1 - 2.33T + 83T^{2} \)
89 \( 1 + 6.49T + 89T^{2} \)
97 \( 1 + 11.1T + 97T^{2} \)
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

−8.521853825300800356300960154555, −8.148976368488367304306496104017, −6.95987313571699561017782970192, −6.45312379275205947103850240902, −5.52986872457028045266353440027, −4.81342366929224163900343308290, −4.10092153936733546476931790192, −2.38699397031891033297111064598, −1.53005830994111875806609654317, −0.801227226310391983901949883436, 0.801227226310391983901949883436, 1.53005830994111875806609654317, 2.38699397031891033297111064598, 4.10092153936733546476931790192, 4.81342366929224163900343308290, 5.52986872457028045266353440027, 6.45312379275205947103850240902, 6.95987313571699561017782970192, 8.148976368488367304306496104017, 8.521853825300800356300960154555

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