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  − 0.432·2-s − 1.81·4-s + 4.09·5-s + 7-s + 1.64·8-s − 1.77·10-s − 2.19·11-s − 2.36·13-s − 0.432·14-s + 2.91·16-s + 2.91·17-s + 6.73·19-s − 7.43·20-s + 0.948·22-s − 4.56·23-s + 11.8·25-s + 1.02·26-s − 1.81·28-s + 4.17·29-s − 0.485·31-s − 4.55·32-s − 1.25·34-s + 4.09·35-s + 5.48·37-s − 2.90·38-s + 6.75·40-s − 10.5·41-s + ⋯
L(s)  = 1  − 0.305·2-s − 0.906·4-s + 1.83·5-s + 0.377·7-s + 0.582·8-s − 0.560·10-s − 0.661·11-s − 0.656·13-s − 0.115·14-s + 0.728·16-s + 0.705·17-s + 1.54·19-s − 1.66·20-s + 0.202·22-s − 0.951·23-s + 2.36·25-s + 0.200·26-s − 0.342·28-s + 0.774·29-s − 0.0871·31-s − 0.805·32-s − 0.215·34-s + 0.693·35-s + 0.902·37-s − 0.471·38-s + 1.06·40-s − 1.65·41-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  =  $(2,\ 8001,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.189493092$
$L(\frac12)$  $\approx$  $2.189493092$
$L(\frac{3}{2})$   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 \( 1 + 0.432T + 2T^{2} \)
5 \( 1 - 4.09T + 5T^{2} \)
11 \( 1 + 2.19T + 11T^{2} \)
13 \( 1 + 2.36T + 13T^{2} \)
17 \( 1 - 2.91T + 17T^{2} \)
19 \( 1 - 6.73T + 19T^{2} \)
23 \( 1 + 4.56T + 23T^{2} \)
29 \( 1 - 4.17T + 29T^{2} \)
31 \( 1 + 0.485T + 31T^{2} \)
37 \( 1 - 5.48T + 37T^{2} \)
41 \( 1 + 10.5T + 41T^{2} \)
43 \( 1 + 6.67T + 43T^{2} \)
47 \( 1 - 10.8T + 47T^{2} \)
53 \( 1 - 9.15T + 53T^{2} \)
59 \( 1 - 1.66T + 59T^{2} \)
61 \( 1 + 14.4T + 61T^{2} \)
67 \( 1 - 14.4T + 67T^{2} \)
71 \( 1 + 2.63T + 71T^{2} \)
73 \( 1 + 0.610T + 73T^{2} \)
79 \( 1 + 7.36T + 79T^{2} \)
83 \( 1 - 11.0T + 83T^{2} \)
89 \( 1 - 7.36T + 89T^{2} \)
97 \( 1 - 10.8T + 97T^{2} \)
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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

−7.88554526411390793766220996727, −7.29369148372728469456770857913, −6.32328683439029544524670309416, −5.46117250202940403195995429719, −5.29271424736132670507333528885, −4.55981225188016285173039167711, −3.38801338383382777575095993931, −2.52885053860601469340282193023, −1.67325969676634930592525304773, −0.816021382635084012554089575550, 0.816021382635084012554089575550, 1.67325969676634930592525304773, 2.52885053860601469340282193023, 3.38801338383382777575095993931, 4.55981225188016285173039167711, 5.29271424736132670507333528885, 5.46117250202940403195995429719, 6.32328683439029544524670309416, 7.29369148372728469456770857913, 7.88554526411390793766220996727

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