Properties

Degree 2
Conductor $ 3 \cdot 7 \cdot 127 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 2.09·2-s + 3-s + 2.40·4-s − 3.29·5-s + 2.09·6-s + 7-s + 0.858·8-s + 9-s − 6.90·10-s − 3.62·11-s + 2.40·12-s − 2.16·13-s + 2.09·14-s − 3.29·15-s − 3.01·16-s + 2.19·17-s + 2.09·18-s − 6.66·19-s − 7.92·20-s + 21-s − 7.61·22-s + 2.36·23-s + 0.858·24-s + 5.82·25-s − 4.53·26-s + 27-s + 2.40·28-s + ⋯
L(s)  = 1  + 1.48·2-s + 0.577·3-s + 1.20·4-s − 1.47·5-s + 0.857·6-s + 0.377·7-s + 0.303·8-s + 0.333·9-s − 2.18·10-s − 1.09·11-s + 0.695·12-s − 0.599·13-s + 0.561·14-s − 0.849·15-s − 0.753·16-s + 0.531·17-s + 0.494·18-s − 1.52·19-s − 1.77·20-s + 0.218·21-s − 1.62·22-s + 0.493·23-s + 0.175·24-s + 1.16·25-s − 0.889·26-s + 0.192·27-s + 0.455·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(2667\)    =    \(3 \cdot 7 \cdot 127\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{2667} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 2667,\ (\ :1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   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 $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 - T \)
7 \( 1 - T \)
127 \( 1 + T \)
good2 \( 1 - 2.09T + 2T^{2} \)
5 \( 1 + 3.29T + 5T^{2} \)
11 \( 1 + 3.62T + 11T^{2} \)
13 \( 1 + 2.16T + 13T^{2} \)
17 \( 1 - 2.19T + 17T^{2} \)
19 \( 1 + 6.66T + 19T^{2} \)
23 \( 1 - 2.36T + 23T^{2} \)
29 \( 1 - 2.06T + 29T^{2} \)
31 \( 1 + 2.19T + 31T^{2} \)
37 \( 1 + 10.1T + 37T^{2} \)
41 \( 1 + 0.647T + 41T^{2} \)
43 \( 1 - 3.67T + 43T^{2} \)
47 \( 1 + 8.59T + 47T^{2} \)
53 \( 1 - 2.84T + 53T^{2} \)
59 \( 1 + 0.592T + 59T^{2} \)
61 \( 1 + 13.5T + 61T^{2} \)
67 \( 1 - 0.664T + 67T^{2} \)
71 \( 1 + 5.65T + 71T^{2} \)
73 \( 1 + 3.24T + 73T^{2} \)
79 \( 1 - 4.01T + 79T^{2} \)
83 \( 1 + 3.40T + 83T^{2} \)
89 \( 1 - 6.31T + 89T^{2} \)
97 \( 1 + 0.0689T + 97T^{2} \)
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

−8.330393042250060725274878315276, −7.55700246370403466597088486816, −7.04166882073594139322080390504, −5.97910815232961638707404265214, −4.88874203030223447246842407383, −4.59881785531194680942793839419, −3.64286368313824415727375722646, −3.06076565460433692811602790337, −2.08995788913651934432165487272, 0, 2.08995788913651934432165487272, 3.06076565460433692811602790337, 3.64286368313824415727375722646, 4.59881785531194680942793839419, 4.88874203030223447246842407383, 5.97910815232961638707404265214, 7.04166882073594139322080390504, 7.55700246370403466597088486816, 8.330393042250060725274878315276

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