Properties

Degree 2
Conductor $ 2 \cdot 7 \cdot 769 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3·3-s + 4-s − 5-s − 3·6-s + 7-s + 8-s + 6·9-s − 10-s + 5·11-s − 3·12-s + 14-s + 3·15-s + 16-s + 4·17-s + 6·18-s − 19-s − 20-s − 3·21-s + 5·22-s − 4·23-s − 3·24-s − 4·25-s − 9·27-s + 28-s + 2·29-s + 3·30-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.73·3-s + 1/2·4-s − 0.447·5-s − 1.22·6-s + 0.377·7-s + 0.353·8-s + 2·9-s − 0.316·10-s + 1.50·11-s − 0.866·12-s + 0.267·14-s + 0.774·15-s + 1/4·16-s + 0.970·17-s + 1.41·18-s − 0.229·19-s − 0.223·20-s − 0.654·21-s + 1.06·22-s − 0.834·23-s − 0.612·24-s − 4/5·25-s − 1.73·27-s + 0.188·28-s + 0.371·29-s + 0.547·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10766 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10766 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(10766\)    =    \(2 \cdot 7 \cdot 769\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{10766} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 10766,\ (\ :1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;7,\;769\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;7,\;769\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - T \)
7 \( 1 - T \)
769 \( 1 + T \)
good3 \( 1 + p T + p T^{2} \)
5 \( 1 + T + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 8 T + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
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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

−16.78078336170323, −16.34287028861824, −15.88425599774677, −14.91616045850167, −14.74884929484108, −13.97352474004152, −13.24360300735766, −12.51929706913999, −12.09213820874948, −11.72373216161625, −11.18168558782512, −10.83613057569982, −9.815446455410439, −9.575403624931581, −8.314836615553331, −7.698474871643559, −6.955505139000391, −6.387928546221555, −5.900446716789503, −5.270381458999585, −4.564018413856995, −4.001507024434619, −3.381466726244980, −1.835578288166316, −1.220584019169284, 0, 1.220584019169284, 1.835578288166316, 3.381466726244980, 4.001507024434619, 4.564018413856995, 5.270381458999585, 5.900446716789503, 6.387928546221555, 6.955505139000391, 7.698474871643559, 8.314836615553331, 9.575403624931581, 9.815446455410439, 10.83613057569982, 11.18168558782512, 11.72373216161625, 12.09213820874948, 12.51929706913999, 13.24360300735766, 13.97352474004152, 14.74884929484108, 14.91616045850167, 15.88425599774677, 16.34287028861824, 16.78078336170323

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