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 − 2.26·3-s + 4-s + 3.87·5-s + 2.26·6-s + 1.72·7-s − 8-s + 2.13·9-s − 3.87·10-s − 0.520·11-s − 2.26·12-s − 3.00·13-s − 1.72·14-s − 8.78·15-s + 16-s − 2.56·17-s − 2.13·18-s − 2.36·19-s + 3.87·20-s − 3.91·21-s + 0.520·22-s − 23-s + 2.26·24-s + 10.0·25-s + 3.00·26-s + 1.95·27-s + 1.72·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.30·3-s + 0.5·4-s + 1.73·5-s + 0.925·6-s + 0.652·7-s − 0.353·8-s + 0.712·9-s − 1.22·10-s − 0.156·11-s − 0.654·12-s − 0.834·13-s − 0.461·14-s − 2.26·15-s + 0.250·16-s − 0.621·17-s − 0.503·18-s − 0.543·19-s + 0.866·20-s − 0.854·21-s + 0.110·22-s − 0.208·23-s + 0.462·24-s + 2.00·25-s + 0.590·26-s + 0.376·27-s + 0.326·28-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 \( 1 + 2.26T + 3T^{2} \)
5 \( 1 - 3.87T + 5T^{2} \)
7 \( 1 - 1.72T + 7T^{2} \)
11 \( 1 + 0.520T + 11T^{2} \)
13 \( 1 + 3.00T + 13T^{2} \)
17 \( 1 + 2.56T + 17T^{2} \)
19 \( 1 + 2.36T + 19T^{2} \)
29 \( 1 + 4.33T + 29T^{2} \)
31 \( 1 - 4.96T + 31T^{2} \)
37 \( 1 + 3.95T + 37T^{2} \)
41 \( 1 - 6.89T + 41T^{2} \)
43 \( 1 - 0.963T + 43T^{2} \)
47 \( 1 + 5.02T + 47T^{2} \)
53 \( 1 - 11.3T + 53T^{2} \)
59 \( 1 + 7.77T + 59T^{2} \)
61 \( 1 + 3.89T + 61T^{2} \)
67 \( 1 + 1.15T + 67T^{2} \)
71 \( 1 + 10.7T + 71T^{2} \)
73 \( 1 + 11.5T + 73T^{2} \)
79 \( 1 + 6.96T + 79T^{2} \)
83 \( 1 - 0.507T + 83T^{2} \)
89 \( 1 - 0.862T + 89T^{2} \)
97 \( 1 - 17.8T + 97T^{2} \)
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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

−7.60010235022454790465404486357, −6.87984657978619911504924824158, −6.18329034956380129435583830553, −5.77508988079383602602040811112, −5.05297181559342955321703511577, −4.45988504895179407433396704271, −2.78059565361489251845495088328, −2.04171577226470200850764841567, −1.27574645118003034296692171154, 0, 1.27574645118003034296692171154, 2.04171577226470200850764841567, 2.78059565361489251845495088328, 4.45988504895179407433396704271, 5.05297181559342955321703511577, 5.77508988079383602602040811112, 6.18329034956380129435583830553, 6.87984657978619911504924824158, 7.60010235022454790465404486357

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