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  + 0.305·2-s − 1.90·4-s + 0.276·5-s + 7-s − 1.19·8-s + 0.0845·10-s + 5.49·11-s + 5.85·13-s + 0.305·14-s + 3.44·16-s − 1.15·17-s − 6.10·19-s − 0.527·20-s + 1.68·22-s − 4.90·23-s − 4.92·25-s + 1.78·26-s − 1.90·28-s − 5.37·29-s − 9.23·31-s + 3.44·32-s − 0.351·34-s + 0.276·35-s + 3.98·37-s − 1.86·38-s − 0.330·40-s − 1.75·41-s + ⋯
L(s)  = 1  + 0.216·2-s − 0.953·4-s + 0.123·5-s + 0.377·7-s − 0.422·8-s + 0.0267·10-s + 1.65·11-s + 1.62·13-s + 0.0817·14-s + 0.861·16-s − 0.279·17-s − 1.40·19-s − 0.117·20-s + 0.358·22-s − 1.02·23-s − 0.984·25-s + 0.350·26-s − 0.360·28-s − 0.998·29-s − 1.65·31-s + 0.608·32-s − 0.0603·34-s + 0.0467·35-s + 0.654·37-s − 0.302·38-s − 0.0522·40-s − 0.274·41-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 \( 1 - 0.305T + 2T^{2} \)
5 \( 1 - 0.276T + 5T^{2} \)
11 \( 1 - 5.49T + 11T^{2} \)
13 \( 1 - 5.85T + 13T^{2} \)
17 \( 1 + 1.15T + 17T^{2} \)
19 \( 1 + 6.10T + 19T^{2} \)
23 \( 1 + 4.90T + 23T^{2} \)
29 \( 1 + 5.37T + 29T^{2} \)
31 \( 1 + 9.23T + 31T^{2} \)
37 \( 1 - 3.98T + 37T^{2} \)
41 \( 1 + 1.75T + 41T^{2} \)
43 \( 1 + 10.6T + 43T^{2} \)
47 \( 1 - 2.53T + 47T^{2} \)
53 \( 1 - 6.75T + 53T^{2} \)
59 \( 1 + 10.1T + 59T^{2} \)
61 \( 1 - 6.36T + 61T^{2} \)
67 \( 1 - 3.74T + 67T^{2} \)
71 \( 1 + 10.2T + 71T^{2} \)
73 \( 1 - 0.769T + 73T^{2} \)
79 \( 1 + 0.396T + 79T^{2} \)
83 \( 1 - 2.27T + 83T^{2} \)
89 \( 1 - 6.61T + 89T^{2} \)
97 \( 1 - 6.50T + 97T^{2} \)
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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

−7.59844220283137356331789426191, −6.53732909436691117384142773147, −6.09131052274994686914929923871, −5.48818259050663494198515566724, −4.44148136243333163896167982399, −3.81333987611695757396088434384, −3.65592556268546187203367784494, −1.99665350770640841284099339829, −1.35138407809896135602577995870, 0, 1.35138407809896135602577995870, 1.99665350770640841284099339829, 3.65592556268546187203367784494, 3.81333987611695757396088434384, 4.44148136243333163896167982399, 5.48818259050663494198515566724, 6.09131052274994686914929923871, 6.53732909436691117384142773147, 7.59844220283137356331789426191

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