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  + 1.96·2-s + 1.87·4-s − 2.35·5-s − 7-s − 0.254·8-s − 4.62·10-s + 2.89·11-s + 6.85·13-s − 1.96·14-s − 4.24·16-s − 2.39·17-s − 4.06·19-s − 4.39·20-s + 5.68·22-s + 2.32·23-s + 0.527·25-s + 13.4·26-s − 1.87·28-s − 3.14·29-s − 6.35·31-s − 7.83·32-s − 4.72·34-s + 2.35·35-s + 5.72·37-s − 7.99·38-s + 0.597·40-s − 5.44·41-s + ⋯
L(s)  = 1  + 1.39·2-s + 0.935·4-s − 1.05·5-s − 0.377·7-s − 0.0898·8-s − 1.46·10-s + 0.871·11-s + 1.90·13-s − 0.525·14-s − 1.06·16-s − 0.582·17-s − 0.932·19-s − 0.983·20-s + 1.21·22-s + 0.485·23-s + 0.105·25-s + 2.64·26-s − 0.353·28-s − 0.583·29-s − 1.14·31-s − 1.38·32-s − 0.809·34-s + 0.397·35-s + 0.941·37-s − 1.29·38-s + 0.0945·40-s − 0.850·41-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 \( 1 - 1.96T + 2T^{2} \)
5 \( 1 + 2.35T + 5T^{2} \)
11 \( 1 - 2.89T + 11T^{2} \)
13 \( 1 - 6.85T + 13T^{2} \)
17 \( 1 + 2.39T + 17T^{2} \)
19 \( 1 + 4.06T + 19T^{2} \)
23 \( 1 - 2.32T + 23T^{2} \)
29 \( 1 + 3.14T + 29T^{2} \)
31 \( 1 + 6.35T + 31T^{2} \)
37 \( 1 - 5.72T + 37T^{2} \)
41 \( 1 + 5.44T + 41T^{2} \)
43 \( 1 + 4.92T + 43T^{2} \)
47 \( 1 - 6.42T + 47T^{2} \)
53 \( 1 - 13.2T + 53T^{2} \)
59 \( 1 - 4.10T + 59T^{2} \)
61 \( 1 - 10.0T + 61T^{2} \)
67 \( 1 + 9.78T + 67T^{2} \)
71 \( 1 - 0.218T + 71T^{2} \)
73 \( 1 + 12.3T + 73T^{2} \)
79 \( 1 + 2.97T + 79T^{2} \)
83 \( 1 + 13.0T + 83T^{2} \)
89 \( 1 + 7.20T + 89T^{2} \)
97 \( 1 + 18.5T + 97T^{2} \)
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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

−7.07434587648783584617363843935, −6.74082922830440185667898273111, −5.94703930184384305160018229317, −5.45832853219841290884146030208, −4.22276918495888612872326753132, −4.03300686725763147236240817836, −3.54362047505118342680729175169, −2.61173860363575556394213836867, −1.41046763309321373277220694462, 0, 1.41046763309321373277220694462, 2.61173860363575556394213836867, 3.54362047505118342680729175169, 4.03300686725763147236240817836, 4.22276918495888612872326753132, 5.45832853219841290884146030208, 5.94703930184384305160018229317, 6.74082922830440185667898273111, 7.07434587648783584617363843935

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