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  − 0.421·2-s − 2.05·3-s − 1.82·4-s + 5-s + 0.867·6-s − 0.0580·7-s + 1.61·8-s + 1.22·9-s − 0.421·10-s + 11-s + 3.74·12-s + 6.32·13-s + 0.0244·14-s − 2.05·15-s + 2.96·16-s + 5.86·17-s − 0.517·18-s + 0.796·19-s − 1.82·20-s + 0.119·21-s − 0.421·22-s + 8.29·23-s − 3.31·24-s + 25-s − 2.67·26-s + 3.64·27-s + 0.105·28-s + ⋯
L(s)  = 1  − 0.298·2-s − 1.18·3-s − 0.910·4-s + 0.447·5-s + 0.354·6-s − 0.0219·7-s + 0.570·8-s + 0.408·9-s − 0.133·10-s + 0.301·11-s + 1.08·12-s + 1.75·13-s + 0.00654·14-s − 0.530·15-s + 0.740·16-s + 1.42·17-s − 0.122·18-s + 0.182·19-s − 0.407·20-s + 0.0260·21-s − 0.0899·22-s + 1.72·23-s − 0.676·24-s + 0.200·25-s − 0.523·26-s + 0.701·27-s + 0.0199·28-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  =  $(2,\ 4015,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.136978308$
$L(\frac12)$  $\approx$  $1.136978308$
$L(\frac{3}{2})$   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 \( 1 + 0.421T + 2T^{2} \)
3 \( 1 + 2.05T + 3T^{2} \)
7 \( 1 + 0.0580T + 7T^{2} \)
13 \( 1 - 6.32T + 13T^{2} \)
17 \( 1 - 5.86T + 17T^{2} \)
19 \( 1 - 0.796T + 19T^{2} \)
23 \( 1 - 8.29T + 23T^{2} \)
29 \( 1 + 1.17T + 29T^{2} \)
31 \( 1 + 0.271T + 31T^{2} \)
37 \( 1 - 3.31T + 37T^{2} \)
41 \( 1 - 1.75T + 41T^{2} \)
43 \( 1 + 1.18T + 43T^{2} \)
47 \( 1 + 11.5T + 47T^{2} \)
53 \( 1 - 3.80T + 53T^{2} \)
59 \( 1 - 1.80T + 59T^{2} \)
61 \( 1 - 7.32T + 61T^{2} \)
67 \( 1 - 11.5T + 67T^{2} \)
71 \( 1 + 2.43T + 71T^{2} \)
79 \( 1 - 10.1T + 79T^{2} \)
83 \( 1 - 1.69T + 83T^{2} \)
89 \( 1 - 10.5T + 89T^{2} \)
97 \( 1 + 17.3T + 97T^{2} \)
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

−8.506356970204294796912252330706, −7.85945448780914204940967131178, −6.75277956721107580222305245692, −6.17460441904799674332588821383, −5.35101720402953611232436462034, −5.04738396052923583135640769846, −3.89247158385796821597474129936, −3.17057299503461533566649008282, −1.35740555163515518867588572215, −0.812569580372492757723131307187, 0.812569580372492757723131307187, 1.35740555163515518867588572215, 3.17057299503461533566649008282, 3.89247158385796821597474129936, 5.04738396052923583135640769846, 5.35101720402953611232436462034, 6.17460441904799674332588821383, 6.75277956721107580222305245692, 7.85945448780914204940967131178, 8.506356970204294796912252330706

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