Properties

Degree 2
Conductor $ 7 \cdot 11 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s − 2·4-s − 5-s − 7-s + 6·9-s − 11-s + 6·12-s − 4·13-s + 3·15-s + 4·16-s + 2·17-s − 6·19-s + 2·20-s + 3·21-s − 5·23-s − 4·25-s − 9·27-s + 2·28-s + 10·29-s + 31-s + 3·33-s + 35-s − 12·36-s − 5·37-s + 12·39-s − 2·41-s − 8·43-s + ⋯
L(s)  = 1  − 1.73·3-s − 4-s − 0.447·5-s − 0.377·7-s + 2·9-s − 0.301·11-s + 1.73·12-s − 1.10·13-s + 0.774·15-s + 16-s + 0.485·17-s − 1.37·19-s + 0.447·20-s + 0.654·21-s − 1.04·23-s − 4/5·25-s − 1.73·27-s + 0.377·28-s + 1.85·29-s + 0.179·31-s + 0.522·33-s + 0.169·35-s − 2·36-s − 0.821·37-s + 1.92·39-s − 0.312·41-s − 1.21·43-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(77\)    =    \(7 \cdot 11\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{77} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 77,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{7,\;11\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{7,\;11\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad7 \( 1 + T \)
11 \( 1 + T \)
good2 \( 1 + p T^{2} \)
3 \( 1 + p T + p T^{2} \)
5 \( 1 + T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 5 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 - T + p T^{2} \)
37 \( 1 + 5 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 3 T + p T^{2} \)
71 \( 1 - T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 6 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 15 T + p T^{2} \)
97 \( 1 + 5 T + p T^{2} \)
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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

−19.56126692589293, −18.79592964745168, −17.74843363531321, −17.19984519791578, −16.28495257845359, −15.20431084275141, −13.75798057830490, −12.41538070383591, −12.09579448828259, −10.54462273276050, −9.842751379499007, −8.090125574816451, −6.567092748055369, −5.301702450605604, −4.260207872588436, 0, 4.260207872588436, 5.301702450605604, 6.567092748055369, 8.090125574816451, 9.842751379499007, 10.54462273276050, 12.09579448828259, 12.41538070383591, 13.75798057830490, 15.20431084275141, 16.28495257845359, 17.19984519791578, 17.74843363531321, 18.79592964745168, 19.56126692589293

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