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  − 2.44·2-s + 3.98·4-s + 1.74·5-s − 7-s − 4.85·8-s − 4.26·10-s − 0.135·11-s − 2.02·13-s + 2.44·14-s + 3.90·16-s + 0.659·17-s − 3.26·19-s + 6.94·20-s + 0.332·22-s − 2.00·23-s − 1.96·25-s + 4.95·26-s − 3.98·28-s − 6.31·29-s + 6.78·31-s + 0.149·32-s − 1.61·34-s − 1.74·35-s + 2.08·37-s + 7.98·38-s − 8.46·40-s + 2.96·41-s + ⋯
L(s)  = 1  − 1.72·2-s + 1.99·4-s + 0.779·5-s − 0.377·7-s − 1.71·8-s − 1.34·10-s − 0.0409·11-s − 0.561·13-s + 0.653·14-s + 0.977·16-s + 0.159·17-s − 0.748·19-s + 1.55·20-s + 0.0708·22-s − 0.418·23-s − 0.392·25-s + 0.971·26-s − 0.753·28-s − 1.17·29-s + 1.21·31-s + 0.0263·32-s − 0.276·34-s − 0.294·35-s + 0.342·37-s + 1.29·38-s − 1.33·40-s + 0.462·41-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 \( 1 + 2.44T + 2T^{2} \)
5 \( 1 - 1.74T + 5T^{2} \)
11 \( 1 + 0.135T + 11T^{2} \)
13 \( 1 + 2.02T + 13T^{2} \)
17 \( 1 - 0.659T + 17T^{2} \)
19 \( 1 + 3.26T + 19T^{2} \)
23 \( 1 + 2.00T + 23T^{2} \)
29 \( 1 + 6.31T + 29T^{2} \)
31 \( 1 - 6.78T + 31T^{2} \)
37 \( 1 - 2.08T + 37T^{2} \)
41 \( 1 - 2.96T + 41T^{2} \)
43 \( 1 + 2.48T + 43T^{2} \)
47 \( 1 - 12.8T + 47T^{2} \)
53 \( 1 - 1.28T + 53T^{2} \)
59 \( 1 - 12.3T + 59T^{2} \)
61 \( 1 + 11.9T + 61T^{2} \)
67 \( 1 + 13.6T + 67T^{2} \)
71 \( 1 + 0.817T + 71T^{2} \)
73 \( 1 - 10.1T + 73T^{2} \)
79 \( 1 - 3.94T + 79T^{2} \)
83 \( 1 - 17.3T + 83T^{2} \)
89 \( 1 - 12.4T + 89T^{2} \)
97 \( 1 - 4.98T + 97T^{2} \)
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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

−7.63886427283233587527840297052, −7.03496782404681014202392193068, −6.22685736626763762234388834376, −5.84508452298041423338354149257, −4.74842480242133052734904372503, −3.70961721605014886384236989970, −2.49914940271903528293757360265, −2.14295803003165685800572509361, −1.07980143346012740124731621700, 0, 1.07980143346012740124731621700, 2.14295803003165685800572509361, 2.49914940271903528293757360265, 3.70961721605014886384236989970, 4.74842480242133052734904372503, 5.84508452298041423338354149257, 6.22685736626763762234388834376, 7.03496782404681014202392193068, 7.63886427283233587527840297052

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