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  + 2.40·2-s + 3.77·4-s + 0.666·5-s + 7-s + 4.25·8-s + 1.60·10-s + 5.05·11-s + 5.76·13-s + 2.40·14-s + 2.68·16-s + 0.396·17-s + 3.56·19-s + 2.51·20-s + 12.1·22-s + 4.24·23-s − 4.55·25-s + 13.8·26-s + 3.77·28-s − 0.519·29-s + 3.84·31-s − 2.06·32-s + 0.952·34-s + 0.666·35-s − 7.00·37-s + 8.57·38-s + 2.83·40-s − 3.06·41-s + ⋯
L(s)  = 1  + 1.69·2-s + 1.88·4-s + 0.298·5-s + 0.377·7-s + 1.50·8-s + 0.506·10-s + 1.52·11-s + 1.60·13-s + 0.642·14-s + 0.670·16-s + 0.0961·17-s + 0.818·19-s + 0.562·20-s + 2.58·22-s + 0.884·23-s − 0.911·25-s + 2.71·26-s + 0.712·28-s − 0.0965·29-s + 0.690·31-s − 0.365·32-s + 0.163·34-s + 0.112·35-s − 1.15·37-s + 1.39·38-s + 0.448·40-s − 0.479·41-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  =  $(2,\ 8001,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $8.214979944$
$L(\frac12)$  $\approx$  $8.214979944$
$L(\frac{3}{2})$   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 \( 1 - 2.40T + 2T^{2} \)
5 \( 1 - 0.666T + 5T^{2} \)
11 \( 1 - 5.05T + 11T^{2} \)
13 \( 1 - 5.76T + 13T^{2} \)
17 \( 1 - 0.396T + 17T^{2} \)
19 \( 1 - 3.56T + 19T^{2} \)
23 \( 1 - 4.24T + 23T^{2} \)
29 \( 1 + 0.519T + 29T^{2} \)
31 \( 1 - 3.84T + 31T^{2} \)
37 \( 1 + 7.00T + 37T^{2} \)
41 \( 1 + 3.06T + 41T^{2} \)
43 \( 1 + 4.79T + 43T^{2} \)
47 \( 1 + 11.2T + 47T^{2} \)
53 \( 1 - 1.29T + 53T^{2} \)
59 \( 1 - 1.96T + 59T^{2} \)
61 \( 1 + 9.45T + 61T^{2} \)
67 \( 1 + 8.88T + 67T^{2} \)
71 \( 1 - 8.51T + 71T^{2} \)
73 \( 1 + 6.50T + 73T^{2} \)
79 \( 1 - 14.8T + 79T^{2} \)
83 \( 1 + 13.6T + 83T^{2} \)
89 \( 1 - 12.6T + 89T^{2} \)
97 \( 1 - 9.21T + 97T^{2} \)
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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

−7.56965758400123316826975253666, −6.65678658190042488559105347950, −6.39886004104044680890076803313, −5.65894922387849866611508512410, −5.01150169070710782104547802994, −4.26955642297587379507685694949, −3.54034856902392427247847265423, −3.17371158144119946530302661912, −1.83807406616482213668882010054, −1.27817136371281216808926895726, 1.27817136371281216808926895726, 1.83807406616482213668882010054, 3.17371158144119946530302661912, 3.54034856902392427247847265423, 4.26955642297587379507685694949, 5.01150169070710782104547802994, 5.65894922387849866611508512410, 6.39886004104044680890076803313, 6.65678658190042488559105347950, 7.56965758400123316826975253666

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