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  − 1.33·2-s − 0.210·4-s + 0.660·5-s + 7-s + 2.95·8-s − 0.883·10-s + 2.50·11-s + 3.59·13-s − 1.33·14-s − 3.53·16-s − 0.647·17-s + 4.90·19-s − 0.139·20-s − 3.34·22-s − 4.27·23-s − 4.56·25-s − 4.81·26-s − 0.210·28-s + 7.60·29-s − 2.73·31-s − 1.18·32-s + 0.866·34-s + 0.660·35-s + 0.693·37-s − 6.55·38-s + 1.95·40-s + 11.1·41-s + ⋯
L(s)  = 1  − 0.945·2-s − 0.105·4-s + 0.295·5-s + 0.377·7-s + 1.04·8-s − 0.279·10-s + 0.754·11-s + 0.997·13-s − 0.357·14-s − 0.883·16-s − 0.157·17-s + 1.12·19-s − 0.0311·20-s − 0.713·22-s − 0.892·23-s − 0.912·25-s − 0.943·26-s − 0.0398·28-s + 1.41·29-s − 0.491·31-s − 0.209·32-s + 0.148·34-s + 0.111·35-s + 0.114·37-s − 1.06·38-s + 0.308·40-s + 1.74·41-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  =  $(2,\ 8001,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.469874113$
$L(\frac12)$  $\approx$  $1.469874113$
$L(\frac{3}{2})$   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 \( 1 + 1.33T + 2T^{2} \)
5 \( 1 - 0.660T + 5T^{2} \)
11 \( 1 - 2.50T + 11T^{2} \)
13 \( 1 - 3.59T + 13T^{2} \)
17 \( 1 + 0.647T + 17T^{2} \)
19 \( 1 - 4.90T + 19T^{2} \)
23 \( 1 + 4.27T + 23T^{2} \)
29 \( 1 - 7.60T + 29T^{2} \)
31 \( 1 + 2.73T + 31T^{2} \)
37 \( 1 - 0.693T + 37T^{2} \)
41 \( 1 - 11.1T + 41T^{2} \)
43 \( 1 + 1.57T + 43T^{2} \)
47 \( 1 - 6.59T + 47T^{2} \)
53 \( 1 + 7.27T + 53T^{2} \)
59 \( 1 - 9.88T + 59T^{2} \)
61 \( 1 - 0.382T + 61T^{2} \)
67 \( 1 + 4.37T + 67T^{2} \)
71 \( 1 - 13.1T + 71T^{2} \)
73 \( 1 - 4.43T + 73T^{2} \)
79 \( 1 + 6.44T + 79T^{2} \)
83 \( 1 - 4.82T + 83T^{2} \)
89 \( 1 - 9.63T + 89T^{2} \)
97 \( 1 + 0.182T + 97T^{2} \)
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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

−7.977452955902796860619264098330, −7.39586384208226808665513826167, −6.51361215726355859577773902263, −5.87026845681061961782315213762, −5.06273757741818102653092849425, −4.18707327403329687870219549011, −3.65748017128087543311595755010, −2.38018814326814476900044126691, −1.45946134405110426953703795188, −0.78058242816577649985403663605, 0.78058242816577649985403663605, 1.45946134405110426953703795188, 2.38018814326814476900044126691, 3.65748017128087543311595755010, 4.18707327403329687870219549011, 5.06273757741818102653092849425, 5.87026845681061961782315213762, 6.51361215726355859577773902263, 7.39586384208226808665513826167, 7.977452955902796860619264098330

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