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 + 0.489·3-s + 4-s + 2.70·5-s − 0.489·6-s + 1.75·7-s − 8-s − 2.76·9-s − 2.70·10-s + 1.86·11-s + 0.489·12-s − 5.77·13-s − 1.75·14-s + 1.32·15-s + 16-s + 0.411·17-s + 2.76·18-s − 3.30·19-s + 2.70·20-s + 0.859·21-s − 1.86·22-s + 23-s − 0.489·24-s + 2.32·25-s + 5.77·26-s − 2.81·27-s + 1.75·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.282·3-s + 0.5·4-s + 1.21·5-s − 0.199·6-s + 0.664·7-s − 0.353·8-s − 0.920·9-s − 0.856·10-s + 0.561·11-s + 0.141·12-s − 1.60·13-s − 0.469·14-s + 0.342·15-s + 0.250·16-s + 0.0997·17-s + 0.650·18-s − 0.758·19-s + 0.605·20-s + 0.187·21-s − 0.397·22-s + 0.208·23-s − 0.0998·24-s + 0.465·25-s + 1.13·26-s − 0.542·27-s + 0.332·28-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 \( 1 - 0.489T + 3T^{2} \)
5 \( 1 - 2.70T + 5T^{2} \)
7 \( 1 - 1.75T + 7T^{2} \)
11 \( 1 - 1.86T + 11T^{2} \)
13 \( 1 + 5.77T + 13T^{2} \)
17 \( 1 - 0.411T + 17T^{2} \)
19 \( 1 + 3.30T + 19T^{2} \)
29 \( 1 + 8.95T + 29T^{2} \)
31 \( 1 - 4.04T + 31T^{2} \)
37 \( 1 - 8.37T + 37T^{2} \)
41 \( 1 + 9.32T + 41T^{2} \)
43 \( 1 + 4.95T + 43T^{2} \)
47 \( 1 + 10.3T + 47T^{2} \)
53 \( 1 - 8.33T + 53T^{2} \)
59 \( 1 - 13.2T + 59T^{2} \)
61 \( 1 - 9.35T + 61T^{2} \)
67 \( 1 - 6.28T + 67T^{2} \)
71 \( 1 + 4.46T + 71T^{2} \)
73 \( 1 + 13.5T + 73T^{2} \)
79 \( 1 + 7.13T + 79T^{2} \)
83 \( 1 - 6.71T + 83T^{2} \)
89 \( 1 - 3.92T + 89T^{2} \)
97 \( 1 + 9.59T + 97T^{2} \)
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

−7.956302115274510132569604908566, −7.00387104960306188711402868615, −6.46039971835462421016513288229, −5.52488734427702859521376534000, −5.13139162535518409750253785169, −3.99710160984130249887223166179, −2.80002033248492676262146920758, −2.21689463997384185224599920175, −1.50918940921191755528648737472, 0, 1.50918940921191755528648737472, 2.21689463997384185224599920175, 2.80002033248492676262146920758, 3.99710160984130249887223166179, 5.13139162535518409750253785169, 5.52488734427702859521376534000, 6.46039971835462421016513288229, 7.00387104960306188711402868615, 7.956302115274510132569604908566

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