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 − 1.04·3-s + 4-s + 3.02·5-s + 1.04·6-s − 3.14·7-s − 8-s − 1.90·9-s − 3.02·10-s + 5.21·11-s − 1.04·12-s − 5.14·13-s + 3.14·14-s − 3.16·15-s + 16-s + 3.04·17-s + 1.90·18-s − 1.27·19-s + 3.02·20-s + 3.29·21-s − 5.21·22-s − 23-s + 1.04·24-s + 4.15·25-s + 5.14·26-s + 5.13·27-s − 3.14·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.604·3-s + 0.5·4-s + 1.35·5-s + 0.427·6-s − 1.19·7-s − 0.353·8-s − 0.634·9-s − 0.956·10-s + 1.57·11-s − 0.302·12-s − 1.42·13-s + 0.841·14-s − 0.817·15-s + 0.250·16-s + 0.738·17-s + 0.448·18-s − 0.292·19-s + 0.676·20-s + 0.719·21-s − 1.11·22-s − 0.208·23-s + 0.213·24-s + 0.830·25-s + 1.00·26-s + 0.987·27-s − 0.595·28-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 \( 1 + 1.04T + 3T^{2} \)
5 \( 1 - 3.02T + 5T^{2} \)
7 \( 1 + 3.14T + 7T^{2} \)
11 \( 1 - 5.21T + 11T^{2} \)
13 \( 1 + 5.14T + 13T^{2} \)
17 \( 1 - 3.04T + 17T^{2} \)
19 \( 1 + 1.27T + 19T^{2} \)
29 \( 1 + 2.37T + 29T^{2} \)
31 \( 1 + 3.86T + 31T^{2} \)
37 \( 1 - 10.8T + 37T^{2} \)
41 \( 1 + 2.37T + 41T^{2} \)
43 \( 1 + 9.52T + 43T^{2} \)
47 \( 1 - 13.1T + 47T^{2} \)
53 \( 1 + 0.167T + 53T^{2} \)
59 \( 1 + 7.58T + 59T^{2} \)
61 \( 1 + 0.958T + 61T^{2} \)
67 \( 1 - 13.0T + 67T^{2} \)
71 \( 1 - 0.625T + 71T^{2} \)
73 \( 1 - 6.70T + 73T^{2} \)
79 \( 1 - 1.23T + 79T^{2} \)
83 \( 1 + 17.6T + 83T^{2} \)
89 \( 1 + 16.6T + 89T^{2} \)
97 \( 1 - 18.5T + 97T^{2} \)
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

−7.61980693237056802824401643975, −6.79372296870516388948101374481, −6.35978869795552754343949674501, −5.80892067435254048790285376617, −5.17785299057573458218785293645, −3.96244653187022124048980240258, −2.95367351760107449313044993537, −2.23315218944133022408527093491, −1.18445567911360158882952250542, 0, 1.18445567911360158882952250542, 2.23315218944133022408527093491, 2.95367351760107449313044993537, 3.96244653187022124048980240258, 5.17785299057573458218785293645, 5.80892067435254048790285376617, 6.35978869795552754343949674501, 6.79372296870516388948101374481, 7.61980693237056802824401643975

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