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 + 3.04·3-s + 4-s − 0.456·5-s − 3.04·6-s + 0.160·7-s − 8-s + 6.25·9-s + 0.456·10-s − 4.62·11-s + 3.04·12-s + 1.12·13-s − 0.160·14-s − 1.38·15-s + 16-s + 1.47·17-s − 6.25·18-s − 4.27·19-s − 0.456·20-s + 0.487·21-s + 4.62·22-s + 23-s − 3.04·24-s − 4.79·25-s − 1.12·26-s + 9.91·27-s + 0.160·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.75·3-s + 0.5·4-s − 0.203·5-s − 1.24·6-s + 0.0605·7-s − 0.353·8-s + 2.08·9-s + 0.144·10-s − 1.39·11-s + 0.878·12-s + 0.311·13-s − 0.0428·14-s − 0.358·15-s + 0.250·16-s + 0.356·17-s − 1.47·18-s − 0.980·19-s − 0.101·20-s + 0.106·21-s + 0.986·22-s + 0.208·23-s − 0.621·24-s − 0.958·25-s − 0.220·26-s + 1.90·27-s + 0.0302·28-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 \( 1 - 3.04T + 3T^{2} \)
5 \( 1 + 0.456T + 5T^{2} \)
7 \( 1 - 0.160T + 7T^{2} \)
11 \( 1 + 4.62T + 11T^{2} \)
13 \( 1 - 1.12T + 13T^{2} \)
17 \( 1 - 1.47T + 17T^{2} \)
19 \( 1 + 4.27T + 19T^{2} \)
29 \( 1 + 4.46T + 29T^{2} \)
31 \( 1 + 10.1T + 31T^{2} \)
37 \( 1 - 3.62T + 37T^{2} \)
41 \( 1 - 5.48T + 41T^{2} \)
43 \( 1 + 12.1T + 43T^{2} \)
47 \( 1 + 9.95T + 47T^{2} \)
53 \( 1 + 5.67T + 53T^{2} \)
59 \( 1 + 2.69T + 59T^{2} \)
61 \( 1 - 1.37T + 61T^{2} \)
67 \( 1 - 11.2T + 67T^{2} \)
71 \( 1 - 0.790T + 71T^{2} \)
73 \( 1 + 6.46T + 73T^{2} \)
79 \( 1 - 2.55T + 79T^{2} \)
83 \( 1 - 2.81T + 83T^{2} \)
89 \( 1 + 8.73T + 89T^{2} \)
97 \( 1 - 12.6T + 97T^{2} \)
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

−7.921539300176314759590220768167, −7.47061748832454245948467387918, −6.63680716064402226670267166032, −5.62699302028826521066198547421, −4.66458661179172382719668540042, −3.64928131063367539426333032460, −3.16469640005246248497397920936, −2.19874109815291023997246132202, −1.69828034101949573932502851923, 0, 1.69828034101949573932502851923, 2.19874109815291023997246132202, 3.16469640005246248497397920936, 3.64928131063367539426333032460, 4.66458661179172382719668540042, 5.62699302028826521066198547421, 6.63680716064402226670267166032, 7.47061748832454245948467387918, 7.921539300176314759590220768167

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