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 + 2·3-s + 4-s − 5-s + 2·6-s + 2·7-s + 8-s + 9-s − 10-s − 5·11-s + 2·12-s − 2·13-s + 2·14-s − 2·15-s + 16-s − 3·17-s + 18-s − 6·19-s − 20-s + 4·21-s − 5·22-s − 23-s + 2·24-s − 4·25-s − 2·26-s − 4·27-s + 2·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.447·5-s + 0.816·6-s + 0.755·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 1.50·11-s + 0.577·12-s − 0.554·13-s + 0.534·14-s − 0.516·15-s + 1/4·16-s − 0.727·17-s + 0.235·18-s − 1.37·19-s − 0.223·20-s + 0.872·21-s − 1.06·22-s − 0.208·23-s + 0.408·24-s − 4/5·25-s − 0.392·26-s − 0.769·27-s + 0.377·28-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 \( 1 - 2 T + p T^{2} \)
5 \( 1 + T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 7 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 + 3 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 + 16 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 16 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−7.69306586890244365183550821206, −7.33136092664972074183455072853, −6.20984871688873478131419909738, −5.46631210375278095549961792991, −4.54663950261094573486040620749, −4.18612527745841938002559454044, −3.11604157674398979015951476785, −2.44338517386560553379649909971, −1.91159709497181123771052394535, 0, 1.91159709497181123771052394535, 2.44338517386560553379649909971, 3.11604157674398979015951476785, 4.18612527745841938002559454044, 4.54663950261094573486040620749, 5.46631210375278095549961792991, 6.20984871688873478131419909738, 7.33136092664972074183455072853, 7.69306586890244365183550821206

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