Properties

Degree 2
Conductor $ 7 \cdot 13 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 2·4-s − 3·5-s − 7-s − 3·9-s + 6·10-s − 6·11-s − 13-s + 2·14-s − 4·16-s + 4·17-s + 6·18-s + 5·19-s − 6·20-s + 12·22-s + 3·23-s + 4·25-s + 2·26-s − 2·28-s − 5·29-s − 3·31-s + 8·32-s − 8·34-s + 3·35-s − 6·36-s − 4·37-s − 10·38-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s − 1.34·5-s − 0.377·7-s − 9-s + 1.89·10-s − 1.80·11-s − 0.277·13-s + 0.534·14-s − 16-s + 0.970·17-s + 1.41·18-s + 1.14·19-s − 1.34·20-s + 2.55·22-s + 0.625·23-s + 4/5·25-s + 0.392·26-s − 0.377·28-s − 0.928·29-s − 0.538·31-s + 1.41·32-s − 1.37·34-s + 0.507·35-s − 36-s − 0.657·37-s − 1.62·38-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(91\)    =    \(7 \cdot 13\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{91} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 91,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{7,\;13\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{7,\;13\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad7 \( 1 + T \)
13 \( 1 + T \)
good2 \( 1 + p T + p T^{2} \)
3 \( 1 + p T^{2} \)
5 \( 1 + 3 T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 + 3 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 - 7 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 6 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 13 T + p T^{2} \)
79 \( 1 - 3 T + p T^{2} \)
83 \( 1 - 15 T + p T^{2} \)
89 \( 1 - 3 T + p T^{2} \)
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

−19.10191848594405, −18.71972589240754, −17.66537031738552, −16.54920901748068, −15.97429303486911, −15.01362508916295, −13.50802048420851, −12.13349961472522, −11.15186582749694, −10.25104185337052, −8.998346772410808, −7.883056466255848, −7.435639297443093, −5.278833137709257, −3.117442763971341, 0, 3.117442763971341, 5.278833137709257, 7.435639297443093, 7.883056466255848, 8.998346772410808, 10.25104185337052, 11.15186582749694, 12.13349961472522, 13.50802048420851, 15.01362508916295, 15.97429303486911, 16.54920901748068, 17.66537031738552, 18.71972589240754, 19.10191848594405

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