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  − 2.49·2-s + 4.22·4-s − 0.373·5-s − 7-s − 5.54·8-s + 0.932·10-s + 6.40·11-s + 2.54·13-s + 2.49·14-s + 5.38·16-s − 4.88·17-s − 5.83·19-s − 1.57·20-s − 15.9·22-s + 8.35·23-s − 4.86·25-s − 6.34·26-s − 4.22·28-s + 1.63·29-s − 6.96·31-s − 2.35·32-s + 12.1·34-s + 0.373·35-s + 7.46·37-s + 14.5·38-s + 2.07·40-s + 9.02·41-s + ⋯
L(s)  = 1  − 1.76·2-s + 2.11·4-s − 0.167·5-s − 0.377·7-s − 1.96·8-s + 0.294·10-s + 1.93·11-s + 0.705·13-s + 0.666·14-s + 1.34·16-s − 1.18·17-s − 1.33·19-s − 0.352·20-s − 3.40·22-s + 1.74·23-s − 0.972·25-s − 1.24·26-s − 0.798·28-s + 0.303·29-s − 1.25·31-s − 0.415·32-s + 2.09·34-s + 0.0631·35-s + 1.22·37-s + 2.36·38-s + 0.327·40-s + 1.40·41-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 \( 1 + 2.49T + 2T^{2} \)
5 \( 1 + 0.373T + 5T^{2} \)
11 \( 1 - 6.40T + 11T^{2} \)
13 \( 1 - 2.54T + 13T^{2} \)
17 \( 1 + 4.88T + 17T^{2} \)
19 \( 1 + 5.83T + 19T^{2} \)
23 \( 1 - 8.35T + 23T^{2} \)
29 \( 1 - 1.63T + 29T^{2} \)
31 \( 1 + 6.96T + 31T^{2} \)
37 \( 1 - 7.46T + 37T^{2} \)
41 \( 1 - 9.02T + 41T^{2} \)
43 \( 1 + 7.64T + 43T^{2} \)
47 \( 1 + 6.14T + 47T^{2} \)
53 \( 1 - 2.53T + 53T^{2} \)
59 \( 1 + 1.55T + 59T^{2} \)
61 \( 1 + 11.0T + 61T^{2} \)
67 \( 1 - 11.4T + 67T^{2} \)
71 \( 1 + 7.04T + 71T^{2} \)
73 \( 1 - 5.49T + 73T^{2} \)
79 \( 1 + 7.21T + 79T^{2} \)
83 \( 1 + 4.98T + 83T^{2} \)
89 \( 1 + 6.29T + 89T^{2} \)
97 \( 1 + 15.3T + 97T^{2} \)
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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

−7.59812970908898895485377680528, −6.71541692473629138659909450092, −6.63203292893840354455706428314, −5.87208673809112509434680885907, −4.44393883238393125821278735787, −3.83539322885809506495770430486, −2.79186717337905710957243148017, −1.81790832717354732237654157519, −1.13223073178406970027683502675, 0, 1.13223073178406970027683502675, 1.81790832717354732237654157519, 2.79186717337905710957243148017, 3.83539322885809506495770430486, 4.44393883238393125821278735787, 5.87208673809112509434680885907, 6.63203292893840354455706428314, 6.71541692473629138659909450092, 7.59812970908898895485377680528

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