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  + 2-s + 2·3-s − 4-s − 2·5-s + 2·6-s − 7-s − 3·8-s + 9-s − 2·10-s + 11-s − 2·12-s + 4·13-s − 14-s − 4·15-s − 16-s + 4·17-s + 18-s + 2·20-s − 2·21-s + 22-s − 4·23-s − 6·24-s − 25-s + 4·26-s − 4·27-s + 28-s − 6·29-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.15·3-s − 1/2·4-s − 0.894·5-s + 0.816·6-s − 0.377·7-s − 1.06·8-s + 1/3·9-s − 0.632·10-s + 0.301·11-s − 0.577·12-s + 1.10·13-s − 0.267·14-s − 1.03·15-s − 1/4·16-s + 0.970·17-s + 0.235·18-s + 0.447·20-s − 0.436·21-s + 0.213·22-s − 0.834·23-s − 1.22·24-s − 1/5·25-s + 0.784·26-s − 0.769·27-s + 0.188·28-s − 1.11·29-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  =  $(2,\ 77,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.325807429$
$L(\frac12)$  $\approx$  $1.325807429$
$L(\frac{3}{2})$   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 \( 1 - T + p T^{2} \)
3 \( 1 - 2 T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 4 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 2 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−19.33714768108551, −18.93790333949262, −17.62486217464210, −16.08313188157145, −15.26279597326968, −14.29047997411069, −13.66117239363331, −12.58384912124745, −11.53310644884625, −9.757325112612361, −8.678915900547505, −7.828675060072816, −5.964439781100220, −4.079308830345754, −3.268883446029810, 3.268883446029810, 4.079308830345754, 5.964439781100220, 7.828675060072816, 8.678915900547505, 9.757325112612361, 11.53310644884625, 12.58384912124745, 13.66117239363331, 14.29047997411069, 15.26279597326968, 16.08313188157145, 17.62486217464210, 18.93790333949262, 19.33714768108551

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