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  + 0.903·2-s − 1.18·4-s − 2.97·5-s − 7-s − 2.87·8-s − 2.69·10-s − 0.876·11-s − 1.42·13-s − 0.903·14-s − 0.235·16-s + 0.579·17-s + 7.77·19-s + 3.52·20-s − 0.792·22-s + 3.51·23-s + 3.87·25-s − 1.28·26-s + 1.18·28-s − 1.75·29-s − 5.47·31-s + 5.54·32-s + 0.524·34-s + 2.97·35-s − 8.19·37-s + 7.02·38-s + 8.57·40-s + 5.68·41-s + ⋯
L(s)  = 1  + 0.639·2-s − 0.591·4-s − 1.33·5-s − 0.377·7-s − 1.01·8-s − 0.851·10-s − 0.264·11-s − 0.395·13-s − 0.241·14-s − 0.0588·16-s + 0.140·17-s + 1.78·19-s + 0.787·20-s − 0.168·22-s + 0.733·23-s + 0.774·25-s − 0.252·26-s + 0.223·28-s − 0.324·29-s − 0.983·31-s + 0.979·32-s + 0.0898·34-s + 0.503·35-s − 1.34·37-s + 1.14·38-s + 1.35·40-s + 0.888·41-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 \( 1 - 0.903T + 2T^{2} \)
5 \( 1 + 2.97T + 5T^{2} \)
11 \( 1 + 0.876T + 11T^{2} \)
13 \( 1 + 1.42T + 13T^{2} \)
17 \( 1 - 0.579T + 17T^{2} \)
19 \( 1 - 7.77T + 19T^{2} \)
23 \( 1 - 3.51T + 23T^{2} \)
29 \( 1 + 1.75T + 29T^{2} \)
31 \( 1 + 5.47T + 31T^{2} \)
37 \( 1 + 8.19T + 37T^{2} \)
41 \( 1 - 5.68T + 41T^{2} \)
43 \( 1 - 7.37T + 43T^{2} \)
47 \( 1 - 11.3T + 47T^{2} \)
53 \( 1 - 7.36T + 53T^{2} \)
59 \( 1 - 8.42T + 59T^{2} \)
61 \( 1 + 14.0T + 61T^{2} \)
67 \( 1 - 1.17T + 67T^{2} \)
71 \( 1 + 10.2T + 71T^{2} \)
73 \( 1 - 11.1T + 73T^{2} \)
79 \( 1 + 1.45T + 79T^{2} \)
83 \( 1 + 17.1T + 83T^{2} \)
89 \( 1 + 0.702T + 89T^{2} \)
97 \( 1 + 17.4T + 97T^{2} \)
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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

−7.33610912429763595847335159394, −7.08449514008224411939104499158, −5.71516552360632455159100143929, −5.43522764963766941941121127547, −4.54355640375116374430478659418, −3.90600105815982688254396739470, −3.33812849566830843606393148902, −2.65907122983687381183261959185, −0.983866491224366985875385250006, 0, 0.983866491224366985875385250006, 2.65907122983687381183261959185, 3.33812849566830843606393148902, 3.90600105815982688254396739470, 4.54355640375116374430478659418, 5.43522764963766941941121127547, 5.71516552360632455159100143929, 7.08449514008224411939104499158, 7.33610912429763595847335159394

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