Properties

Degree 2
Conductor $ 2 \cdot 23 \cdot 131 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 8-s − 3·9-s + 2·11-s − 2·13-s + 16-s − 6·17-s + 3·18-s + 2·19-s − 2·22-s + 23-s − 5·25-s + 2·26-s + 2·29-s − 8·31-s − 32-s + 6·34-s − 3·36-s + 4·37-s − 2·38-s + 6·41-s + 2·43-s + 2·44-s − 46-s − 7·49-s + 5·50-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.353·8-s − 9-s + 0.603·11-s − 0.554·13-s + 1/4·16-s − 1.45·17-s + 0.707·18-s + 0.458·19-s − 0.426·22-s + 0.208·23-s − 25-s + 0.392·26-s + 0.371·29-s − 1.43·31-s − 0.176·32-s + 1.02·34-s − 1/2·36-s + 0.657·37-s − 0.324·38-s + 0.937·41-s + 0.304·43-s + 0.301·44-s − 0.147·46-s − 49-s + 0.707·50-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  =  0
Selberg data  =  $(2,\ 6026,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.8733587421$
$L(\frac12)$  $\approx$  $0.8733587421$
$L(\frac{3}{2})$   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 $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + T \)
23 \( 1 - T \)
131 \( 1 + T \)
good3 \( 1 + p T^{2} \)
5 \( 1 + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 12 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−8.136255743189544970253673972909, −7.45874998251234741861678003789, −6.73820977248035087534882054690, −6.07413394149159065098784541999, −5.35929050554392728258872019661, −4.41659294228689110102232518535, −3.52251830803368942520302850264, −2.57633528057070295510804316455, −1.87419581676770393357237438018, −0.52491069196242228043430119310, 0.52491069196242228043430119310, 1.87419581676770393357237438018, 2.57633528057070295510804316455, 3.52251830803368942520302850264, 4.41659294228689110102232518535, 5.35929050554392728258872019661, 6.07413394149159065098784541999, 6.73820977248035087534882054690, 7.45874998251234741861678003789, 8.136255743189544970253673972909

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