Properties

Degree 2
Conductor $ 17 \cdot 43 $
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  + 1.96·2-s − 1.75·3-s + 1.87·4-s + 0.290·5-s − 3.44·6-s − 2.03·7-s − 0.249·8-s + 0.0632·9-s + 0.571·10-s − 1.62·11-s − 3.27·12-s − 2.15·13-s − 4.01·14-s − 0.508·15-s − 4.23·16-s − 17-s + 0.124·18-s − 3.82·19-s + 0.543·20-s + 3.56·21-s − 3.19·22-s + 4.47·23-s + 0.437·24-s − 4.91·25-s − 4.24·26-s + 5.13·27-s − 3.81·28-s + ⋯
L(s)  = 1  + 1.39·2-s − 1.01·3-s + 0.936·4-s + 0.129·5-s − 1.40·6-s − 0.770·7-s − 0.0883·8-s + 0.0210·9-s + 0.180·10-s − 0.489·11-s − 0.946·12-s − 0.598·13-s − 1.07·14-s − 0.131·15-s − 1.05·16-s − 0.242·17-s + 0.0293·18-s − 0.878·19-s + 0.121·20-s + 0.778·21-s − 0.681·22-s + 0.933·23-s + 0.0892·24-s − 0.983·25-s − 0.832·26-s + 0.989·27-s − 0.721·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(731\)    =    \(17 \cdot 43\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{731} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 731,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{17,\;43\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{17,\;43\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad17 \( 1 + T \)
43 \( 1 + T \)
good2 \( 1 - 1.96T + 2T^{2} \)
3 \( 1 + 1.75T + 3T^{2} \)
5 \( 1 - 0.290T + 5T^{2} \)
7 \( 1 + 2.03T + 7T^{2} \)
11 \( 1 + 1.62T + 11T^{2} \)
13 \( 1 + 2.15T + 13T^{2} \)
19 \( 1 + 3.82T + 19T^{2} \)
23 \( 1 - 4.47T + 23T^{2} \)
29 \( 1 + 3.55T + 29T^{2} \)
31 \( 1 - 9.03T + 31T^{2} \)
37 \( 1 - 4.37T + 37T^{2} \)
41 \( 1 + 10.4T + 41T^{2} \)
47 \( 1 + 10.5T + 47T^{2} \)
53 \( 1 + 2.68T + 53T^{2} \)
59 \( 1 - 8.62T + 59T^{2} \)
61 \( 1 + 4.69T + 61T^{2} \)
67 \( 1 + 11.1T + 67T^{2} \)
71 \( 1 - 9.13T + 71T^{2} \)
73 \( 1 - 16.6T + 73T^{2} \)
79 \( 1 - 12.6T + 79T^{2} \)
83 \( 1 - 11.1T + 83T^{2} \)
89 \( 1 - 7.63T + 89T^{2} \)
97 \( 1 + 13.1T + 97T^{2} \)
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

−10.20249742475880609980218580251, −9.304802053356928375698902842062, −8.071302888123564264638710379393, −6.57919068349132840376503631849, −6.38122556795597730655589975124, −5.26586651723312163148572646294, −4.75710408721870182009411095392, −3.51167641547133325165950826536, −2.46931529185023526413853953681, 0, 2.46931529185023526413853953681, 3.51167641547133325165950826536, 4.75710408721870182009411095392, 5.26586651723312163148572646294, 6.38122556795597730655589975124, 6.57919068349132840376503631849, 8.071302888123564264638710379393, 9.304802053356928375698902842062, 10.20249742475880609980218580251

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