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  − 1.18·2-s − 0.601·4-s − 4.39·5-s + 7-s + 3.07·8-s + 5.20·10-s + 1.66·11-s + 1.04·13-s − 1.18·14-s − 2.43·16-s − 2.31·17-s − 1.96·19-s + 2.64·20-s − 1.96·22-s − 2.53·23-s + 14.3·25-s − 1.23·26-s − 0.601·28-s − 3.94·29-s + 2.35·31-s − 3.27·32-s + 2.73·34-s − 4.39·35-s − 3.81·37-s + 2.32·38-s − 13.5·40-s − 5.25·41-s + ⋯
L(s)  = 1  − 0.836·2-s − 0.300·4-s − 1.96·5-s + 0.377·7-s + 1.08·8-s + 1.64·10-s + 0.501·11-s + 0.288·13-s − 0.316·14-s − 0.609·16-s − 0.561·17-s − 0.450·19-s + 0.591·20-s − 0.419·22-s − 0.528·23-s + 2.86·25-s − 0.241·26-s − 0.113·28-s − 0.733·29-s + 0.423·31-s − 0.578·32-s + 0.469·34-s − 0.743·35-s − 0.627·37-s + 0.376·38-s − 2.13·40-s − 0.820·41-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 \( 1 + 1.18T + 2T^{2} \)
5 \( 1 + 4.39T + 5T^{2} \)
11 \( 1 - 1.66T + 11T^{2} \)
13 \( 1 - 1.04T + 13T^{2} \)
17 \( 1 + 2.31T + 17T^{2} \)
19 \( 1 + 1.96T + 19T^{2} \)
23 \( 1 + 2.53T + 23T^{2} \)
29 \( 1 + 3.94T + 29T^{2} \)
31 \( 1 - 2.35T + 31T^{2} \)
37 \( 1 + 3.81T + 37T^{2} \)
41 \( 1 + 5.25T + 41T^{2} \)
43 \( 1 - 1.58T + 43T^{2} \)
47 \( 1 - 5.23T + 47T^{2} \)
53 \( 1 + 6.13T + 53T^{2} \)
59 \( 1 - 1.48T + 59T^{2} \)
61 \( 1 - 0.959T + 61T^{2} \)
67 \( 1 - 4.40T + 67T^{2} \)
71 \( 1 + 4.45T + 71T^{2} \)
73 \( 1 - 1.66T + 73T^{2} \)
79 \( 1 - 0.363T + 79T^{2} \)
83 \( 1 - 9.34T + 83T^{2} \)
89 \( 1 - 3.44T + 89T^{2} \)
97 \( 1 - 13.6T + 97T^{2} \)
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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

−7.68303241012477671868704694017, −7.12928856872857446910468872526, −6.39546746120966935875445354170, −5.15853195436850417559875363555, −4.46892378294435880594886862300, −3.98473293230687688843046363730, −3.33991375941013502710182245482, −1.96830099579191750490943233214, −0.872644942956393490708974784664, 0, 0.872644942956393490708974784664, 1.96830099579191750490943233214, 3.33991375941013502710182245482, 3.98473293230687688843046363730, 4.46892378294435880594886862300, 5.15853195436850417559875363555, 6.39546746120966935875445354170, 7.12928856872857446910468872526, 7.68303241012477671868704694017

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