Properties

Degree 2
Conductor $ 3 \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  + 0.692·2-s + 3-s − 1.52·4-s − 2.78·5-s + 0.692·6-s + 7-s − 2.43·8-s + 9-s − 1.92·10-s − 0.348·11-s − 1.52·12-s + 2.17·13-s + 0.692·14-s − 2.78·15-s + 1.35·16-s + 7.42·17-s + 0.692·18-s − 0.255·19-s + 4.22·20-s + 21-s − 0.241·22-s − 4.98·23-s − 2.43·24-s + 2.73·25-s + 1.50·26-s + 27-s − 1.52·28-s + ⋯
L(s)  = 1  + 0.489·2-s + 0.577·3-s − 0.760·4-s − 1.24·5-s + 0.282·6-s + 0.377·7-s − 0.861·8-s + 0.333·9-s − 0.608·10-s − 0.105·11-s − 0.438·12-s + 0.603·13-s + 0.185·14-s − 0.718·15-s + 0.338·16-s + 1.80·17-s + 0.163·18-s − 0.0586·19-s + 0.945·20-s + 0.218·21-s − 0.0514·22-s − 1.03·23-s − 0.497·24-s + 0.547·25-s + 0.295·26-s + 0.192·27-s − 0.287·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(2667\)    =    \(3 \cdot 7 \cdot 127\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{2667} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 2667,\ (\ :1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   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 $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 - T \)
7 \( 1 - T \)
127 \( 1 + T \)
good2 \( 1 - 0.692T + 2T^{2} \)
5 \( 1 + 2.78T + 5T^{2} \)
11 \( 1 + 0.348T + 11T^{2} \)
13 \( 1 - 2.17T + 13T^{2} \)
17 \( 1 - 7.42T + 17T^{2} \)
19 \( 1 + 0.255T + 19T^{2} \)
23 \( 1 + 4.98T + 23T^{2} \)
29 \( 1 + 6.88T + 29T^{2} \)
31 \( 1 + 10.1T + 31T^{2} \)
37 \( 1 + 1.64T + 37T^{2} \)
41 \( 1 - 3.56T + 41T^{2} \)
43 \( 1 + 9.34T + 43T^{2} \)
47 \( 1 - 9.73T + 47T^{2} \)
53 \( 1 + 7.44T + 53T^{2} \)
59 \( 1 + 2.01T + 59T^{2} \)
61 \( 1 + 7.69T + 61T^{2} \)
67 \( 1 + 6.52T + 67T^{2} \)
71 \( 1 - 13.7T + 71T^{2} \)
73 \( 1 + 9.37T + 73T^{2} \)
79 \( 1 + 5.92T + 79T^{2} \)
83 \( 1 - 1.68T + 83T^{2} \)
89 \( 1 + 17.6T + 89T^{2} \)
97 \( 1 + 0.216T + 97T^{2} \)
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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

−8.295926829838730603213955931291, −7.86250425310618428354266566388, −7.25001629553438894140287985788, −5.87328138432475504780655383797, −5.30119380499332497139575449503, −4.18486869708857855341031591002, −3.76081292997925957555670404027, −3.11133182178127022360390571296, −1.51945173405302223573002331817, 0, 1.51945173405302223573002331817, 3.11133182178127022360390571296, 3.76081292997925957555670404027, 4.18486869708857855341031591002, 5.30119380499332497139575449503, 5.87328138432475504780655383797, 7.25001629553438894140287985788, 7.86250425310618428354266566388, 8.295926829838730603213955931291

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