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  − 2.18·2-s + 3-s + 2.78·4-s + 1.88·5-s − 2.18·6-s − 4.23·7-s − 1.72·8-s + 9-s − 4.11·10-s − 2.82·11-s + 2.78·12-s − 3.44·13-s + 9.26·14-s + 1.88·15-s − 1.80·16-s − 17-s − 2.18·18-s + 4.03·19-s + 5.24·20-s − 4.23·21-s + 6.18·22-s + 7.09·23-s − 1.72·24-s − 1.45·25-s + 7.54·26-s + 27-s − 11.8·28-s + ⋯
L(s)  = 1  − 1.54·2-s + 0.577·3-s + 1.39·4-s + 0.841·5-s − 0.893·6-s − 1.60·7-s − 0.608·8-s + 0.333·9-s − 1.30·10-s − 0.852·11-s + 0.804·12-s − 0.956·13-s + 2.47·14-s + 0.486·15-s − 0.451·16-s − 0.242·17-s − 0.515·18-s + 0.924·19-s + 1.17·20-s − 0.924·21-s + 1.31·22-s + 1.47·23-s − 0.351·24-s − 0.291·25-s + 1.48·26-s + 0.192·27-s − 2.23·28-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 $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 - T \)
17 \( 1 + T \)
157 \( 1 - T \)
good2 \( 1 + 2.18T + 2T^{2} \)
5 \( 1 - 1.88T + 5T^{2} \)
7 \( 1 + 4.23T + 7T^{2} \)
11 \( 1 + 2.82T + 11T^{2} \)
13 \( 1 + 3.44T + 13T^{2} \)
19 \( 1 - 4.03T + 19T^{2} \)
23 \( 1 - 7.09T + 23T^{2} \)
29 \( 1 - 4.45T + 29T^{2} \)
31 \( 1 + 6.06T + 31T^{2} \)
37 \( 1 - 0.948T + 37T^{2} \)
41 \( 1 + 1.88T + 41T^{2} \)
43 \( 1 - 2.93T + 43T^{2} \)
47 \( 1 - 7.81T + 47T^{2} \)
53 \( 1 + 3.16T + 53T^{2} \)
59 \( 1 + 1.33T + 59T^{2} \)
61 \( 1 - 2.15T + 61T^{2} \)
67 \( 1 + 4.67T + 67T^{2} \)
71 \( 1 - 1.47T + 71T^{2} \)
73 \( 1 - 3.77T + 73T^{2} \)
79 \( 1 - 13.2T + 79T^{2} \)
83 \( 1 - 6.59T + 83T^{2} \)
89 \( 1 + 16.0T + 89T^{2} \)
97 \( 1 - 7.74T + 97T^{2} \)
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

−7.47028618541136584090999728777, −7.11666257466965311423577795731, −6.44177643409391029073421888630, −5.58637930412769233671714875762, −4.78400157902750369135126516374, −3.49264380799499570627556095831, −2.69719048384592717949130006722, −2.27038228582018395980958525070, −1.06274911894937543024746736046, 0, 1.06274911894937543024746736046, 2.27038228582018395980958525070, 2.69719048384592717949130006722, 3.49264380799499570627556095831, 4.78400157902750369135126516374, 5.58637930412769233671714875762, 6.44177643409391029073421888630, 7.11666257466965311423577795731, 7.47028618541136584090999728777

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