Properties

Degree 2
Conductor $ 3 \cdot 17 \cdot 157 $
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  − 1.24·2-s − 3-s − 0.444·4-s − 3.92·5-s + 1.24·6-s − 1.90·7-s + 3.04·8-s + 9-s + 4.89·10-s + 1.97·11-s + 0.444·12-s + 3.57·13-s + 2.37·14-s + 3.92·15-s − 2.91·16-s − 17-s − 1.24·18-s − 5.19·19-s + 1.74·20-s + 1.90·21-s − 2.45·22-s − 3.01·23-s − 3.04·24-s + 10.4·25-s − 4.45·26-s − 27-s + 0.845·28-s + ⋯
L(s)  = 1  − 0.881·2-s − 0.577·3-s − 0.222·4-s − 1.75·5-s + 0.509·6-s − 0.718·7-s + 1.07·8-s + 0.333·9-s + 1.54·10-s + 0.594·11-s + 0.128·12-s + 0.991·13-s + 0.633·14-s + 1.01·15-s − 0.728·16-s − 0.242·17-s − 0.293·18-s − 1.19·19-s + 0.389·20-s + 0.415·21-s − 0.523·22-s − 0.627·23-s − 0.622·24-s + 2.08·25-s − 0.874·26-s − 0.192·27-s + 0.159·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8007\)    =    \(3 \cdot 17 \cdot 157\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{8007} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 8007,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;17,\;157\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;17,\;157\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 + T \)
17 \( 1 + T \)
157 \( 1 + T \)
good2 \( 1 + 1.24T + 2T^{2} \)
5 \( 1 + 3.92T + 5T^{2} \)
7 \( 1 + 1.90T + 7T^{2} \)
11 \( 1 - 1.97T + 11T^{2} \)
13 \( 1 - 3.57T + 13T^{2} \)
19 \( 1 + 5.19T + 19T^{2} \)
23 \( 1 + 3.01T + 23T^{2} \)
29 \( 1 + 3.31T + 29T^{2} \)
31 \( 1 + 2.36T + 31T^{2} \)
37 \( 1 + 6.13T + 37T^{2} \)
41 \( 1 + 5.59T + 41T^{2} \)
43 \( 1 + 0.968T + 43T^{2} \)
47 \( 1 - 10.0T + 47T^{2} \)
53 \( 1 - 9.56T + 53T^{2} \)
59 \( 1 + 5.13T + 59T^{2} \)
61 \( 1 + 0.466T + 61T^{2} \)
67 \( 1 + 2.98T + 67T^{2} \)
71 \( 1 - 2.55T + 71T^{2} \)
73 \( 1 - 11.8T + 73T^{2} \)
79 \( 1 + 1.47T + 79T^{2} \)
83 \( 1 - 17.0T + 83T^{2} \)
89 \( 1 + 5.04T + 89T^{2} \)
97 \( 1 - 10.2T + 97T^{2} \)
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

−7.62192585900616383634257357833, −6.88512838947099360626941752803, −6.41898119007764234637720522134, −5.38364292062196386036630189266, −4.42049262754958559609103887391, −3.92046296783485568251718742455, −3.45606430807740034524885078567, −1.88516298385803381462518745822, −0.73522570622780817045919465050, 0, 0.73522570622780817045919465050, 1.88516298385803381462518745822, 3.45606430807740034524885078567, 3.92046296783485568251718742455, 4.42049262754958559609103887391, 5.38364292062196386036630189266, 6.41898119007764234637720522134, 6.88512838947099360626941752803, 7.62192585900616383634257357833

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