Properties

Degree 2
Conductor $ 3^{2} \cdot 7 \cdot 127 $
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  − 1.35·2-s − 0.156·4-s − 4.06·5-s + 7-s + 2.92·8-s + 5.52·10-s − 4.14·11-s + 3.33·13-s − 1.35·14-s − 3.66·16-s + 1.50·17-s + 5.47·19-s + 0.637·20-s + 5.62·22-s + 6.86·23-s + 11.5·25-s − 4.52·26-s − 0.156·28-s + 9.24·29-s − 7.77·31-s − 0.883·32-s − 2.04·34-s − 4.06·35-s + 8.24·37-s − 7.43·38-s − 11.9·40-s + 4.27·41-s + ⋯
L(s)  = 1  − 0.960·2-s − 0.0783·4-s − 1.81·5-s + 0.377·7-s + 1.03·8-s + 1.74·10-s − 1.24·11-s + 0.924·13-s − 0.362·14-s − 0.915·16-s + 0.364·17-s + 1.25·19-s + 0.142·20-s + 1.19·22-s + 1.43·23-s + 2.30·25-s − 0.887·26-s − 0.0295·28-s + 1.71·29-s − 1.39·31-s − 0.156·32-s − 0.350·34-s − 0.687·35-s + 1.35·37-s − 1.20·38-s − 1.88·40-s + 0.667·41-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8001\)    =    \(3^{2} \cdot 7 \cdot 127\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{8001} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 8001,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.7495751163$
$L(\frac12)$  $\approx$  $0.7495751163$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;7,\;127\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;7,\;127\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 \)
7 \( 1 - T \)
127 \( 1 - T \)
good2 \( 1 + 1.35T + 2T^{2} \)
5 \( 1 + 4.06T + 5T^{2} \)
11 \( 1 + 4.14T + 11T^{2} \)
13 \( 1 - 3.33T + 13T^{2} \)
17 \( 1 - 1.50T + 17T^{2} \)
19 \( 1 - 5.47T + 19T^{2} \)
23 \( 1 - 6.86T + 23T^{2} \)
29 \( 1 - 9.24T + 29T^{2} \)
31 \( 1 + 7.77T + 31T^{2} \)
37 \( 1 - 8.24T + 37T^{2} \)
41 \( 1 - 4.27T + 41T^{2} \)
43 \( 1 + 4.47T + 43T^{2} \)
47 \( 1 - 8.84T + 47T^{2} \)
53 \( 1 + 4.29T + 53T^{2} \)
59 \( 1 + 3.63T + 59T^{2} \)
61 \( 1 + 3.59T + 61T^{2} \)
67 \( 1 + 0.939T + 67T^{2} \)
71 \( 1 + 4.09T + 71T^{2} \)
73 \( 1 - 6.53T + 73T^{2} \)
79 \( 1 + 7.66T + 79T^{2} \)
83 \( 1 - 0.904T + 83T^{2} \)
89 \( 1 - 0.889T + 89T^{2} \)
97 \( 1 + 10.8T + 97T^{2} \)
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

−7.87485140219173594664493044161, −7.49497663018523353479893069395, −6.93308138196333528796179820129, −5.64179025225740409775421724287, −4.86309387775437090670241426622, −4.36675573933317535472201431705, −3.44643301179530350387469046067, −2.80175684119794000268268917291, −1.22254165349620556729641014016, −0.59610553868764021542031292551, 0.59610553868764021542031292551, 1.22254165349620556729641014016, 2.80175684119794000268268917291, 3.44643301179530350387469046067, 4.36675573933317535472201431705, 4.86309387775437090670241426622, 5.64179025225740409775421724287, 6.93308138196333528796179820129, 7.49497663018523353479893069395, 7.87485140219173594664493044161

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