Properties

Degree 2
Conductor $ 2 \cdot 23 \cdot 131 $
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-s + 2.29·3-s + 4-s − 1.62·5-s − 2.29·6-s − 0.601·7-s − 8-s + 2.28·9-s + 1.62·10-s + 0.118·11-s + 2.29·12-s + 4.81·13-s + 0.601·14-s − 3.73·15-s + 16-s − 3.51·17-s − 2.28·18-s − 2.74·19-s − 1.62·20-s − 1.38·21-s − 0.118·22-s − 23-s − 2.29·24-s − 2.36·25-s − 4.81·26-s − 1.64·27-s − 0.601·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.32·3-s + 0.5·4-s − 0.725·5-s − 0.938·6-s − 0.227·7-s − 0.353·8-s + 0.762·9-s + 0.513·10-s + 0.0355·11-s + 0.663·12-s + 1.33·13-s + 0.160·14-s − 0.963·15-s + 0.250·16-s − 0.851·17-s − 0.538·18-s − 0.630·19-s − 0.362·20-s − 0.301·21-s − 0.0251·22-s − 0.208·23-s − 0.469·24-s − 0.473·25-s − 0.944·26-s − 0.315·27-s − 0.113·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6026\)    =    \(2 \cdot 23 \cdot 131\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{6026} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 6026,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;23,\;131\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;23,\;131\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + T \)
23 \( 1 + T \)
131 \( 1 + T \)
good3 \( 1 - 2.29T + 3T^{2} \)
5 \( 1 + 1.62T + 5T^{2} \)
7 \( 1 + 0.601T + 7T^{2} \)
11 \( 1 - 0.118T + 11T^{2} \)
13 \( 1 - 4.81T + 13T^{2} \)
17 \( 1 + 3.51T + 17T^{2} \)
19 \( 1 + 2.74T + 19T^{2} \)
29 \( 1 - 5.80T + 29T^{2} \)
31 \( 1 + 6.64T + 31T^{2} \)
37 \( 1 + 4.44T + 37T^{2} \)
41 \( 1 - 8.02T + 41T^{2} \)
43 \( 1 - 11.2T + 43T^{2} \)
47 \( 1 - 0.815T + 47T^{2} \)
53 \( 1 + 7.82T + 53T^{2} \)
59 \( 1 + 10.7T + 59T^{2} \)
61 \( 1 + 3.23T + 61T^{2} \)
67 \( 1 + 8.38T + 67T^{2} \)
71 \( 1 - 3.85T + 71T^{2} \)
73 \( 1 - 6.09T + 73T^{2} \)
79 \( 1 + 15.8T + 79T^{2} \)
83 \( 1 + 0.448T + 83T^{2} \)
89 \( 1 - 10.4T + 89T^{2} \)
97 \( 1 + 4.65T + 97T^{2} \)
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

−7.900850057643015426584950195250, −7.37707208532100830525200154524, −6.45882578145888538200260426835, −5.87977370553946691264495625836, −4.46160098136207779823828166043, −3.83602517990520039896801101958, −3.16080555859955560252280545854, −2.33075727471395888398090103043, −1.42982173691969568271625287696, 0, 1.42982173691969568271625287696, 2.33075727471395888398090103043, 3.16080555859955560252280545854, 3.83602517990520039896801101958, 4.46160098136207779823828166043, 5.87977370553946691264495625836, 6.45882578145888538200260426835, 7.37707208532100830525200154524, 7.900850057643015426584950195250

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