Properties

Degree 2
Conductor $ 3^{2} \cdot 7 \cdot 127 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.24·2-s + 3.05·4-s + 3.31·5-s + 7-s − 2.37·8-s − 7.45·10-s − 5.47·11-s − 4.39·13-s − 2.24·14-s − 0.778·16-s + 2.99·17-s − 0.499·19-s + 10.1·20-s + 12.3·22-s + 6.47·23-s + 5.98·25-s + 9.88·26-s + 3.05·28-s − 2.28·29-s + 4.20·31-s + 6.49·32-s − 6.73·34-s + 3.31·35-s − 7.84·37-s + 1.12·38-s − 7.85·40-s + 7.43·41-s + ⋯
L(s)  = 1  − 1.58·2-s + 1.52·4-s + 1.48·5-s + 0.377·7-s − 0.838·8-s − 2.35·10-s − 1.65·11-s − 1.21·13-s − 0.600·14-s − 0.194·16-s + 0.726·17-s − 0.114·19-s + 2.26·20-s + 2.62·22-s + 1.34·23-s + 1.19·25-s + 1.93·26-s + 0.577·28-s − 0.423·29-s + 0.755·31-s + 1.14·32-s − 1.15·34-s + 0.560·35-s − 1.28·37-s + 0.182·38-s − 1.24·40-s + 1.16·41-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  =  0
Selberg data  =  $(2,\ 8001,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.061390986$
$L(\frac12)$  $\approx$  $1.061390986$
$L(\frac{3}{2})$   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 \( 1 + 2.24T + 2T^{2} \)
5 \( 1 - 3.31T + 5T^{2} \)
11 \( 1 + 5.47T + 11T^{2} \)
13 \( 1 + 4.39T + 13T^{2} \)
17 \( 1 - 2.99T + 17T^{2} \)
19 \( 1 + 0.499T + 19T^{2} \)
23 \( 1 - 6.47T + 23T^{2} \)
29 \( 1 + 2.28T + 29T^{2} \)
31 \( 1 - 4.20T + 31T^{2} \)
37 \( 1 + 7.84T + 37T^{2} \)
41 \( 1 - 7.43T + 41T^{2} \)
43 \( 1 - 6.74T + 43T^{2} \)
47 \( 1 - 8.99T + 47T^{2} \)
53 \( 1 - 1.35T + 53T^{2} \)
59 \( 1 - 5.28T + 59T^{2} \)
61 \( 1 - 0.533T + 61T^{2} \)
67 \( 1 + 14.0T + 67T^{2} \)
71 \( 1 - 6.62T + 71T^{2} \)
73 \( 1 - 6.36T + 73T^{2} \)
79 \( 1 + 15.8T + 79T^{2} \)
83 \( 1 + 5.96T + 83T^{2} \)
89 \( 1 + 6.57T + 89T^{2} \)
97 \( 1 - 12.5T + 97T^{2} \)
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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

−7.79656308308309796310476181265, −7.41384799050929119616443185695, −6.77291139978178641573905624677, −5.69202771476325915494409779220, −5.34466199076492002878985009164, −4.53198888295063806809425457803, −2.79504580397303400659905375665, −2.48577796021722712250966110597, −1.63888859281150592536915867729, −0.65333581594142916217838837260, 0.65333581594142916217838837260, 1.63888859281150592536915867729, 2.48577796021722712250966110597, 2.79504580397303400659905375665, 4.53198888295063806809425457803, 5.34466199076492002878985009164, 5.69202771476325915494409779220, 6.77291139978178641573905624677, 7.41384799050929119616443185695, 7.79656308308309796310476181265

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