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  + 0.981·2-s − 1.03·4-s − 2.59·5-s + 7-s − 2.98·8-s − 2.54·10-s + 0.832·11-s − 5.10·13-s + 0.981·14-s − 0.855·16-s − 7.10·17-s − 7.54·19-s + 2.68·20-s + 0.817·22-s − 8.33·23-s + 1.73·25-s − 5.01·26-s − 1.03·28-s + 8.62·29-s − 6.02·31-s + 5.12·32-s − 6.97·34-s − 2.59·35-s − 5.52·37-s − 7.41·38-s + 7.73·40-s + 8.68·41-s + ⋯
L(s)  = 1  + 0.694·2-s − 0.517·4-s − 1.16·5-s + 0.377·7-s − 1.05·8-s − 0.805·10-s + 0.250·11-s − 1.41·13-s + 0.262·14-s − 0.213·16-s − 1.72·17-s − 1.73·19-s + 0.600·20-s + 0.174·22-s − 1.73·23-s + 0.346·25-s − 0.983·26-s − 0.195·28-s + 1.60·29-s − 1.08·31-s + 0.905·32-s − 1.19·34-s − 0.438·35-s − 0.908·37-s − 1.20·38-s + 1.22·40-s + 1.35·41-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  =  $(2,\ 8001,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.3149272294$
$L(\frac12)$  $\approx$  $0.3149272294$
$L(\frac{3}{2})$   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 \( 1 - 0.981T + 2T^{2} \)
5 \( 1 + 2.59T + 5T^{2} \)
11 \( 1 - 0.832T + 11T^{2} \)
13 \( 1 + 5.10T + 13T^{2} \)
17 \( 1 + 7.10T + 17T^{2} \)
19 \( 1 + 7.54T + 19T^{2} \)
23 \( 1 + 8.33T + 23T^{2} \)
29 \( 1 - 8.62T + 29T^{2} \)
31 \( 1 + 6.02T + 31T^{2} \)
37 \( 1 + 5.52T + 37T^{2} \)
41 \( 1 - 8.68T + 41T^{2} \)
43 \( 1 - 4.88T + 43T^{2} \)
47 \( 1 + 2.27T + 47T^{2} \)
53 \( 1 + 3.81T + 53T^{2} \)
59 \( 1 - 3.88T + 59T^{2} \)
61 \( 1 + 7.64T + 61T^{2} \)
67 \( 1 - 9.03T + 67T^{2} \)
71 \( 1 - 9.47T + 71T^{2} \)
73 \( 1 + 6.75T + 73T^{2} \)
79 \( 1 - 5.46T + 79T^{2} \)
83 \( 1 + 9.63T + 83T^{2} \)
89 \( 1 + 9.34T + 89T^{2} \)
97 \( 1 + 0.647T + 97T^{2} \)
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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

−8.053304288741150938764225188020, −7.00978911904032282718567788140, −6.48913982299483425508282052802, −5.60838738298573625484102933039, −4.72175678099691787544826287497, −4.18446401856904285490757614077, −4.02733776976251794273010600891, −2.74720710187737299827494781178, −2.04703652506591994651758585025, −0.23378839832751369666253754405, 0.23378839832751369666253754405, 2.04703652506591994651758585025, 2.74720710187737299827494781178, 4.02733776976251794273010600891, 4.18446401856904285490757614077, 4.72175678099691787544826287497, 5.60838738298573625484102933039, 6.48913982299483425508282052802, 7.00978911904032282718567788140, 8.053304288741150938764225188020

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