Properties

Degree 2
Conductor $ 7 \cdot 11 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 2·4-s + 3·5-s + 7-s − 2·9-s − 11-s − 2·12-s − 4·13-s + 3·15-s + 4·16-s − 6·17-s + 2·19-s − 6·20-s + 21-s + 3·23-s + 4·25-s − 5·27-s − 2·28-s − 6·29-s + 5·31-s − 33-s + 3·35-s + 4·36-s + 11·37-s − 4·39-s + 6·41-s + 8·43-s + ⋯
L(s)  = 1  + 0.577·3-s − 4-s + 1.34·5-s + 0.377·7-s − 2/3·9-s − 0.301·11-s − 0.577·12-s − 1.10·13-s + 0.774·15-s + 16-s − 1.45·17-s + 0.458·19-s − 1.34·20-s + 0.218·21-s + 0.625·23-s + 4/5·25-s − 0.962·27-s − 0.377·28-s − 1.11·29-s + 0.898·31-s − 0.174·33-s + 0.507·35-s + 2/3·36-s + 1.80·37-s − 0.640·39-s + 0.937·41-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  =  0
Selberg data  =  $(2,\ 77,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.032606710$
$L(\frac12)$  $\approx$  $1.032606710$
$L(\frac{3}{2})$   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 \( 1 - T + p T^{2} \)
5 \( 1 - 3 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 - 11 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 9 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 - 9 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 3 T + p T^{2} \)
97 \( 1 + T + p T^{2} \)
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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

−19.46062145735827, −18.29346493296987, −17.51988448912480, −16.98191065484000, −15.11977487376336, −14.24843777324879, −13.60450574831141, −12.75123359214434, −11.03928612573384, −9.594904136117516, −9.121939609810074, −7.795436734706540, −5.892377313972436, −4.699402245292434, −2.524896273242881, 2.524896273242881, 4.699402245292434, 5.892377313972436, 7.795436734706540, 9.121939609810074, 9.594904136117516, 11.03928612573384, 12.75123359214434, 13.60450574831141, 14.24843777324879, 15.11977487376336, 16.98191065484000, 17.51988448912480, 18.29346493296987, 19.46062145735827

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