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  + 2.45·2-s + 4.01·4-s − 1.23·5-s + 7-s + 4.92·8-s − 3.03·10-s − 2.19·11-s + 1.52·13-s + 2.45·14-s + 4.06·16-s − 7.06·17-s − 1.45·19-s − 4.96·20-s − 5.38·22-s − 2.60·23-s − 3.46·25-s + 3.73·26-s + 4.01·28-s − 6.11·29-s + 1.83·31-s + 0.101·32-s − 17.3·34-s − 1.23·35-s + 0.914·37-s − 3.56·38-s − 6.10·40-s + 1.13·41-s + ⋯
L(s)  = 1  + 1.73·2-s + 2.00·4-s − 0.554·5-s + 0.377·7-s + 1.74·8-s − 0.960·10-s − 0.662·11-s + 0.422·13-s + 0.655·14-s + 1.01·16-s − 1.71·17-s − 0.333·19-s − 1.11·20-s − 1.14·22-s − 0.544·23-s − 0.692·25-s + 0.732·26-s + 0.757·28-s − 1.13·29-s + 0.329·31-s + 0.0179·32-s − 2.97·34-s − 0.209·35-s + 0.150·37-s − 0.577·38-s − 0.965·40-s + 0.177·41-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 \( 1 - 2.45T + 2T^{2} \)
5 \( 1 + 1.23T + 5T^{2} \)
11 \( 1 + 2.19T + 11T^{2} \)
13 \( 1 - 1.52T + 13T^{2} \)
17 \( 1 + 7.06T + 17T^{2} \)
19 \( 1 + 1.45T + 19T^{2} \)
23 \( 1 + 2.60T + 23T^{2} \)
29 \( 1 + 6.11T + 29T^{2} \)
31 \( 1 - 1.83T + 31T^{2} \)
37 \( 1 - 0.914T + 37T^{2} \)
41 \( 1 - 1.13T + 41T^{2} \)
43 \( 1 - 2.48T + 43T^{2} \)
47 \( 1 + 0.620T + 47T^{2} \)
53 \( 1 + 6.86T + 53T^{2} \)
59 \( 1 + 8.13T + 59T^{2} \)
61 \( 1 - 5.09T + 61T^{2} \)
67 \( 1 + 5.28T + 67T^{2} \)
71 \( 1 + 4.94T + 71T^{2} \)
73 \( 1 - 3.24T + 73T^{2} \)
79 \( 1 + 8.62T + 79T^{2} \)
83 \( 1 - 14.6T + 83T^{2} \)
89 \( 1 + 2.62T + 89T^{2} \)
97 \( 1 - 4.23T + 97T^{2} \)
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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

−7.36145430199954052244984908848, −6.50735530837628379782415455250, −6.01060419801298238925684065705, −5.26232972800414523653214220763, −4.49433630020797767515863894216, −4.12617297435819589623554219809, −3.34877473446048063578097169806, −2.44096605029810718843551568927, −1.80312502875150527816709895254, 0, 1.80312502875150527816709895254, 2.44096605029810718843551568927, 3.34877473446048063578097169806, 4.12617297435819589623554219809, 4.49433630020797767515863894216, 5.26232972800414523653214220763, 6.01060419801298238925684065705, 6.50735530837628379782415455250, 7.36145430199954052244984908848

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