Properties

Degree 2
Conductor $ 3^{2} \cdot 7 \cdot 127 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.423·2-s − 1.82·4-s − 2.66·5-s + 7-s − 1.61·8-s − 1.12·10-s − 3.97·11-s + 4.95·13-s + 0.423·14-s + 2.95·16-s − 5.63·17-s + 3.72·19-s + 4.84·20-s − 1.68·22-s + 4.13·23-s + 2.08·25-s + 2.09·26-s − 1.82·28-s − 7.04·29-s − 2.57·31-s + 4.48·32-s − 2.38·34-s − 2.66·35-s + 2.65·37-s + 1.57·38-s + 4.30·40-s − 2.05·41-s + ⋯
L(s)  = 1  + 0.299·2-s − 0.910·4-s − 1.19·5-s + 0.377·7-s − 0.572·8-s − 0.356·10-s − 1.19·11-s + 1.37·13-s + 0.113·14-s + 0.738·16-s − 1.36·17-s + 0.853·19-s + 1.08·20-s − 0.359·22-s + 0.862·23-s + 0.417·25-s + 0.411·26-s − 0.344·28-s − 1.30·29-s − 0.462·31-s + 0.793·32-s − 0.409·34-s − 0.449·35-s + 0.436·37-s + 0.255·38-s + 0.681·40-s − 0.320·41-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8001\)    =    \(3^{2} \cdot 7 \cdot 127\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{8001} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 8001,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;7,\;127\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;7,\;127\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 \)
7 \( 1 - T \)
127 \( 1 + T \)
good2 \( 1 - 0.423T + 2T^{2} \)
5 \( 1 + 2.66T + 5T^{2} \)
11 \( 1 + 3.97T + 11T^{2} \)
13 \( 1 - 4.95T + 13T^{2} \)
17 \( 1 + 5.63T + 17T^{2} \)
19 \( 1 - 3.72T + 19T^{2} \)
23 \( 1 - 4.13T + 23T^{2} \)
29 \( 1 + 7.04T + 29T^{2} \)
31 \( 1 + 2.57T + 31T^{2} \)
37 \( 1 - 2.65T + 37T^{2} \)
41 \( 1 + 2.05T + 41T^{2} \)
43 \( 1 - 4.03T + 43T^{2} \)
47 \( 1 - 7.52T + 47T^{2} \)
53 \( 1 + 0.214T + 53T^{2} \)
59 \( 1 - 12.5T + 59T^{2} \)
61 \( 1 + 2.37T + 61T^{2} \)
67 \( 1 - 4.26T + 67T^{2} \)
71 \( 1 - 13.7T + 71T^{2} \)
73 \( 1 + 10.8T + 73T^{2} \)
79 \( 1 - 2.69T + 79T^{2} \)
83 \( 1 + 4.22T + 83T^{2} \)
89 \( 1 + 4.08T + 89T^{2} \)
97 \( 1 - 0.135T + 97T^{2} \)
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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

−7.63057903616563733447900770492, −6.94375755762361057427502476933, −5.85670247121540844038520137631, −5.30935230384992017194843232308, −4.58883805755193896040424345629, −3.90207004276416229248917259307, −3.43224634244624169113734790385, −2.39682580938357416322277817963, −0.991614986287796923906064892692, 0, 0.991614986287796923906064892692, 2.39682580938357416322277817963, 3.43224634244624169113734790385, 3.90207004276416229248917259307, 4.58883805755193896040424345629, 5.30935230384992017194843232308, 5.85670247121540844038520137631, 6.94375755762361057427502476933, 7.63057903616563733447900770492

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