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 − 2.99·3-s + 4-s − 3.60·5-s + 2.99·6-s + 0.0770·7-s − 8-s + 5.94·9-s + 3.60·10-s − 3.44·11-s − 2.99·12-s − 1.64·13-s − 0.0770·14-s + 10.7·15-s + 16-s + 1.47·17-s − 5.94·18-s − 4.21·19-s − 3.60·20-s − 0.230·21-s + 3.44·22-s − 23-s + 2.99·24-s + 7.96·25-s + 1.64·26-s − 8.79·27-s + 0.0770·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.72·3-s + 0.5·4-s − 1.61·5-s + 1.22·6-s + 0.0291·7-s − 0.353·8-s + 1.98·9-s + 1.13·10-s − 1.03·11-s − 0.863·12-s − 0.454·13-s − 0.0205·14-s + 2.78·15-s + 0.250·16-s + 0.357·17-s − 1.40·18-s − 0.967·19-s − 0.805·20-s − 0.0502·21-s + 0.734·22-s − 0.208·23-s + 0.610·24-s + 1.59·25-s + 0.321·26-s − 1.69·27-s + 0.0145·28-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 \( 1 + 2.99T + 3T^{2} \)
5 \( 1 + 3.60T + 5T^{2} \)
7 \( 1 - 0.0770T + 7T^{2} \)
11 \( 1 + 3.44T + 11T^{2} \)
13 \( 1 + 1.64T + 13T^{2} \)
17 \( 1 - 1.47T + 17T^{2} \)
19 \( 1 + 4.21T + 19T^{2} \)
29 \( 1 - 1.09T + 29T^{2} \)
31 \( 1 + 7.91T + 31T^{2} \)
37 \( 1 + 5.70T + 37T^{2} \)
41 \( 1 - 4.71T + 41T^{2} \)
43 \( 1 + 7.51T + 43T^{2} \)
47 \( 1 - 12.4T + 47T^{2} \)
53 \( 1 - 3.15T + 53T^{2} \)
59 \( 1 - 6.93T + 59T^{2} \)
61 \( 1 - 1.16T + 61T^{2} \)
67 \( 1 + 7.07T + 67T^{2} \)
71 \( 1 + 11.7T + 71T^{2} \)
73 \( 1 - 11.7T + 73T^{2} \)
79 \( 1 - 9.37T + 79T^{2} \)
83 \( 1 + 1.82T + 83T^{2} \)
89 \( 1 + 5.14T + 89T^{2} \)
97 \( 1 - 9.78T + 97T^{2} \)
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

−7.48998406758065514728538329888, −7.24666347512330026105695806121, −6.40242205786589364317375441932, −5.58295561542327229555472333981, −4.93680388152435224606464062010, −4.21386339227520988370858039116, −3.34655807784161983342776576697, −2.03993019900064092348418328871, −0.66752973816918681782109332342, 0, 0.66752973816918681782109332342, 2.03993019900064092348418328871, 3.34655807784161983342776576697, 4.21386339227520988370858039116, 4.93680388152435224606464062010, 5.58295561542327229555472333981, 6.40242205786589364317375441932, 7.24666347512330026105695806121, 7.48998406758065514728538329888

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