Properties

Degree 2
Conductor $ 2 \cdot 23 \cdot 131 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 0.672·3-s + 4-s − 2.38·5-s + 0.672·6-s − 0.506·7-s − 8-s − 2.54·9-s + 2.38·10-s + 1.82·11-s − 0.672·12-s + 1.61·13-s + 0.506·14-s + 1.60·15-s + 16-s − 5.88·17-s + 2.54·18-s − 3.78·19-s − 2.38·20-s + 0.340·21-s − 1.82·22-s − 23-s + 0.672·24-s + 0.681·25-s − 1.61·26-s + 3.73·27-s − 0.506·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.388·3-s + 0.5·4-s − 1.06·5-s + 0.274·6-s − 0.191·7-s − 0.353·8-s − 0.849·9-s + 0.753·10-s + 0.549·11-s − 0.194·12-s + 0.446·13-s + 0.135·14-s + 0.414·15-s + 0.250·16-s − 1.42·17-s + 0.600·18-s − 0.868·19-s − 0.532·20-s + 0.0743·21-s − 0.388·22-s − 0.208·23-s + 0.137·24-s + 0.136·25-s − 0.316·26-s + 0.718·27-s − 0.0957·28-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 \( 1 + 0.672T + 3T^{2} \)
5 \( 1 + 2.38T + 5T^{2} \)
7 \( 1 + 0.506T + 7T^{2} \)
11 \( 1 - 1.82T + 11T^{2} \)
13 \( 1 - 1.61T + 13T^{2} \)
17 \( 1 + 5.88T + 17T^{2} \)
19 \( 1 + 3.78T + 19T^{2} \)
29 \( 1 - 9.57T + 29T^{2} \)
31 \( 1 - 4.60T + 31T^{2} \)
37 \( 1 - 8.91T + 37T^{2} \)
41 \( 1 - 11.6T + 41T^{2} \)
43 \( 1 + 6.39T + 43T^{2} \)
47 \( 1 + 1.18T + 47T^{2} \)
53 \( 1 - 2.70T + 53T^{2} \)
59 \( 1 - 8.49T + 59T^{2} \)
61 \( 1 + 9.41T + 61T^{2} \)
67 \( 1 - 9.70T + 67T^{2} \)
71 \( 1 + 7.20T + 71T^{2} \)
73 \( 1 - 0.843T + 73T^{2} \)
79 \( 1 + 13.7T + 79T^{2} \)
83 \( 1 - 3.54T + 83T^{2} \)
89 \( 1 + 8.85T + 89T^{2} \)
97 \( 1 - 16.5T + 97T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.018074787046628885112357723308, −6.93235533708935784698738228345, −6.42656628864834642585585908518, −5.90638591185530534239294171738, −4.60087067389798947884020150671, −4.17643161875427757657309103448, −3.09352172978608081022867790454, −2.33387962916475461815415695420, −0.938800171414450475986711813746, 0, 0.938800171414450475986711813746, 2.33387962916475461815415695420, 3.09352172978608081022867790454, 4.17643161875427757657309103448, 4.60087067389798947884020150671, 5.90638591185530534239294171738, 6.42656628864834642585585908518, 6.93235533708935784698738228345, 8.018074787046628885112357723308

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