Properties

Degree 2
Conductor $ 3^{2} \cdot 5 \cdot 89 $
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  − 1.77·2-s + 1.14·4-s + 5-s − 3.19·7-s + 1.52·8-s − 1.77·10-s + 2.70·11-s + 2.03·13-s + 5.65·14-s − 4.98·16-s − 2.67·17-s + 1.45·19-s + 1.14·20-s − 4.79·22-s + 5.62·23-s + 25-s − 3.60·26-s − 3.63·28-s − 3.44·29-s − 8.62·31-s + 5.78·32-s + 4.73·34-s − 3.19·35-s + 9.46·37-s − 2.58·38-s + 1.52·40-s + 10.1·41-s + ⋯
L(s)  = 1  − 1.25·2-s + 0.570·4-s + 0.447·5-s − 1.20·7-s + 0.538·8-s − 0.560·10-s + 0.815·11-s + 0.564·13-s + 1.51·14-s − 1.24·16-s − 0.648·17-s + 0.334·19-s + 0.255·20-s − 1.02·22-s + 1.17·23-s + 0.200·25-s − 0.707·26-s − 0.687·28-s − 0.638·29-s − 1.54·31-s + 1.02·32-s + 0.812·34-s − 0.539·35-s + 1.55·37-s − 0.419·38-s + 0.240·40-s + 1.57·41-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4005\)    =    \(3^{2} \cdot 5 \cdot 89\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{4005} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 4005,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.8465077058$
$L(\frac12)$  $\approx$  $0.8465077058$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;5,\;89\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;5,\;89\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 \)
5 \( 1 - T \)
89 \( 1 - T \)
good2 \( 1 + 1.77T + 2T^{2} \)
7 \( 1 + 3.19T + 7T^{2} \)
11 \( 1 - 2.70T + 11T^{2} \)
13 \( 1 - 2.03T + 13T^{2} \)
17 \( 1 + 2.67T + 17T^{2} \)
19 \( 1 - 1.45T + 19T^{2} \)
23 \( 1 - 5.62T + 23T^{2} \)
29 \( 1 + 3.44T + 29T^{2} \)
31 \( 1 + 8.62T + 31T^{2} \)
37 \( 1 - 9.46T + 37T^{2} \)
41 \( 1 - 10.1T + 41T^{2} \)
43 \( 1 - 2.51T + 43T^{2} \)
47 \( 1 + 1.05T + 47T^{2} \)
53 \( 1 - 1.72T + 53T^{2} \)
59 \( 1 - 3.39T + 59T^{2} \)
61 \( 1 + 14.9T + 61T^{2} \)
67 \( 1 + 2.16T + 67T^{2} \)
71 \( 1 + 6.88T + 71T^{2} \)
73 \( 1 + 8.79T + 73T^{2} \)
79 \( 1 - 5.97T + 79T^{2} \)
83 \( 1 - 7.29T + 83T^{2} \)
97 \( 1 - 15.6T + 97T^{2} \)
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

−8.868988090363139298126077619834, −7.69897294738430165510829767391, −7.16025730757129420272312956415, −6.38369279014473729909921120157, −5.81397996285198339730365471142, −4.60535526190545172852769565753, −3.74315434451885671100319853688, −2.75892051525672126965771585104, −1.65072544106115250859749613367, −0.66100454836990368912986947045, 0.66100454836990368912986947045, 1.65072544106115250859749613367, 2.75892051525672126965771585104, 3.74315434451885671100319853688, 4.60535526190545172852769565753, 5.81397996285198339730365471142, 6.38369279014473729909921120157, 7.16025730757129420272312956415, 7.69897294738430165510829767391, 8.868988090363139298126077619834

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