Properties

Degree 2
Conductor $ 3^{2} \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 + 2.40·4-s + 3.29·5-s + 7-s − 0.858·8-s − 6.90·10-s + 3.62·11-s − 2.16·13-s − 2.09·14-s − 3.01·16-s − 2.19·17-s − 6.66·19-s + 7.92·20-s − 7.61·22-s − 2.36·23-s + 5.82·25-s + 4.53·26-s + 2.40·28-s − 2.06·29-s − 2.19·31-s + 8.04·32-s + 4.60·34-s + 3.29·35-s − 10.1·37-s + 13.9·38-s − 2.82·40-s + 0.647·41-s + ⋯
L(s)  = 1  − 1.48·2-s + 1.20·4-s + 1.47·5-s + 0.377·7-s − 0.303·8-s − 2.18·10-s + 1.09·11-s − 0.599·13-s − 0.561·14-s − 0.753·16-s − 0.531·17-s − 1.52·19-s + 1.77·20-s − 1.62·22-s − 0.493·23-s + 1.16·25-s + 0.889·26-s + 0.455·28-s − 0.382·29-s − 0.393·31-s + 1.42·32-s + 0.789·34-s + 0.556·35-s − 1.66·37-s + 2.27·38-s − 0.446·40-s + 0.101·41-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  =  \(1\)
Selberg data  =  \((2,\ 8001,\ (\ :1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   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 \( 1 \)
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

−7.59023652776967271973834405214, −6.83176275880193191824582727212, −6.38760912046798310469491953870, −5.63854095132407408674556076604, −4.71658492410925193406748774582, −3.93995324189305407984113147747, −2.48843884456712230385783681781, −1.94105546729738287151163311347, −1.35544145584558757428642471806, 0, 1.35544145584558757428642471806, 1.94105546729738287151163311347, 2.48843884456712230385783681781, 3.93995324189305407984113147747, 4.71658492410925193406748774582, 5.63854095132407408674556076604, 6.38760912046798310469491953870, 6.83176275880193191824582727212, 7.59023652776967271973834405214

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